A229052 -id:A229052 - cp's OEIS frontend

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 15, 1, 1, 4, 70, 220, 1, 1, 5, 210, 5005, 4845, 1, 1, 6, 495, 48620, 735471, 142506, 1, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1

Offset: 0

Paul D. Hanna, Sep 04 2013

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Paul D. Hanna, Rows 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

{T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

Original entry on oeis.org

1, 2, 7, 1, 3, 4, 1, 5, 2, 2, 1, 8, 9, 0, 1, 5, 2, 2, 5, 2, 2, 2, 3, 8, 2, 5, 7, 8, 7, 9, 0, 9, 3, 5, 6, 2, 4, 9, 7, 6, 8, 6, 4, 9, 8, 7, 7, 1, 7, 6, 2, 6, 7, 0, 1, 1, 6, 4, 4, 1, 0, 8, 0, 1, 6, 9, 7, 4, 7, 7, 5, 8, 5, 6, 6, 5, 5, 3, 0, 7, 5, 0, 6, 2, 3, 9, 3

Offset: 1

Vaclav Kotesovec, Nov 05 2012

			1.2713415221890152252223825787909356249768649877176...
For comparison, sqrt(Pi/2) = 1.2533141373155002512078826424055226265034933703050...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Dedekind Eta Function
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A195907, A069998, A167009, A167010, A229052.

RealDigits[Sum[E^(-2*k^2), {k,-Infinity,Infinity}], 10, 200][[1]]
RealDigits[EllipticTheta[3,0,1/E^2],10,200][[1]] (* Vaclav Kotesovec, Sep 22 2013 *)

1 + 2*suminf(n=1, exp(-2*n^2)) \\ Charles R Greathouse IV, Jun 06 2016

(eta(2*I/Pi))^5 / (eta(I/Pi)^2 * eta(4*I/Pi)^2) \\ Jianing Song, Oct 13 2021

Equals Jacobi theta_{3}(0,exp(-2)). - G. C. Greubel, Feb 01 2017

Equals eta(2*i/Pi)^5 / (eta(i/Pi)*eta(4*i/Pi))^2, where eta(t) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + ... is the Dedekind eta function without the q^(1/24) factor in powers of q = exp(2*Pi*i*t) (Cf. A000122). - Jianing Song, Oct 14 2021

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A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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A218792 Decimal expansion of Sum_{n = -oo..oo} e^(-2*n^2).

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