The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A323182

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A323182

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.
(history; published version)

#47 by Alois P. Heinz at Wed Mar 03 12:16:46 EST 2021

STATUS

proposed

approved

#46 by Seiichi Manyama at Wed Mar 03 12:07:19 EST 2021

STATUS

editing

proposed

#45 by Seiichi Manyama at Wed Mar 03 12:03:35 EST 2021

CROSSREFS

Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

#44 by Seiichi Manyama at Wed Mar 03 11:57:20 EST 2021

FORMULA

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

#43 by Seiichi Manyama at Wed Mar 03 11:55:56 EST 2021

NAME

Square array AT(n,k), n >= 0, k >= 0, read by antidiagonals, where AT(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

FORMULA

AT(0,k) = 1, AT(1,k) = 2 * k and AT(n,k) = 2 * k * AT(n-1,k) - AT(n-2,k) for n > 1.

#42 by Seiichi Manyama at Wed Mar 03 11:52:17 EST 2021

EXAMPLE

1, 1, 1, 1, 1, 1, 1, ...

0, 2, 4, 6, 8, 10, 12, ...

-1, 3, 15, 35, 63, 99, 143, ...

0, 4, 56, 204, 496, 980, 1704, ...

1, 5, 209, 1189, 3905, 9701, 20305, ...

0, 6, 780, 6930, 30744, 96030, 241956, ...

-1, 7, 2911, 40391, 242047, 950599, 2883167, ...

#41 by Seiichi Manyama at Wed Mar 03 11:50:06 EST 2021

PROG

(PARI) T(n, m k) = sum(kj=0, n, (2*mk-2)^kj*binomial(n+1+k, j, 2*kj+1)); \\ Seiichi Manyama, Mar 03 2021

#40 by Seiichi Manyama at Wed Mar 03 11:49:10 EST 2021

PROG

(PARI) T(n, m) = sum(k=0, n, (2*m-2)^k*binomial(n+1+k, 2*k+1)); \\ _Seiichi Manyama_, Mar 03 2021

#39 by Seiichi Manyama at Wed Mar 03 11:48:11 EST 2021

PROG

(PARI) T(n, m) = sum(k=0, n, (2*m-2)^k*binomial(n+1+k, 2*k+1));

STATUS

approved

editing

#38 by Alois P. Heinz at Tue Jan 08 08:08:06 EST 2019

STATUS

proposed

approved