A218792 -id:A218792 - cp's OEIS frontend

A167010 a(n) = Sum_{k=0..n} C(n,k)^n.

1, 2, 6, 56, 1810, 206252, 86874564, 132282417920, 770670360699138, 16425660314368351892, 1367610300690018553312276, 419460465362069257397304825200, 509571049488109525160616367158261124, 2290638298071684282149128235413262383804352

Offset: 0

Paul D. Hanna, Nov 17 2009

The number of n*n 0-1 matrices with equal numbers of nonzeros in every row. - David Eppstein, Jan 19 2012

			The triangle A209427 of coefficients C(n,k)^n, n>=k>=0, begins:
  1;
  1,     1;
  1,     4,        1;
  1,    27,       27,        1;
  1,   256,     1296,      256,        1;
  1,  3125,   100000,   100000,     3125,     1;
  1, 46656, 11390625, 64000000, 11390625, 46656,    1; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..59
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A000312, A014062, A066300, A167009, A167007, A209427, A218792.

[(&+[Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n, k]^n, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2012 *)

PARI
```
a(n)=sum(k=0,n,binomial(n,k)^n)
```

[sum(binomial(n,j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167007. [Paul D. Hanna, Nov 18 2009]
If n is even then a(n) ~ c * exp(-1/4) * 2^(n^2 + n/2)/((Pi*n)^(n/2)), where c = Sum_{k = -oo..oo} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^2. - Ilya Gutkovskiy, Jul 15 2020

A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

Original entry on oeis.org

1, 2, 8, 170, 16512, 6643782, 11582386286, 79450506979090, 2334899414608412672, 265166261617029717011822, 128442558588779813655233443038, 238431997806538515396060130910954852

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			The triangle A209330 of coefficients C(n^2, n*k), n>=k>=0, begins:
  1;
  1,       1;
  1,       6,          1;
  1,      84,         84,          1;
  1,    1820,      12870,       1820,          1;
  1,   53130,    3268760,    3268760,      53130,       1;
  1, 1947792, 1251677700, 9075135300, 1251677700, 1947792,     1; ...
in which the row sums form this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A014062, A167006, A167010, A209330, A218792, A306846.

[(&+[Binomial(n^2, n*j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Table[Sum[Binomial[n^2,n*k],{k,0,n}],{n,0,15}] (* Harvey P. Dale, Dec 11 2011 *)

PARI
```
a(n)=sum(k=0,n,binomial(n^2,n*k))
```

[sum(binomial(n^2, n*j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Ignoring initial term, equals the logarithmic derivative of A167006. - Paul D. Hanna, Nov 18 2009
If n is even then a(n) ~ c * 2^(n^2 + 1/2)/(n*sqrt(Pi)), where c = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... (see A218792). - Vaclav Kotesovec, Nov 05 2012
If n is odd then c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... - Vaclav Kotesovec, Nov 06 2012
a(n) = A306846(n^2,n) = [x^(n^2)] (1-x)^(n-1)/((1-x)^n - x^n) for n > 0. - Seiichi Manyama, Oct 11 2021

A328812 Constant term in the expansion of (Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 2, 10, 164, 9826, 2031252, 1622278624, 4579408029576, 51207103076632066, 2052124795850957537060, 330463219813679264204224300, 192454957455454582636391397662856, 454577215426865313388106323928590128736, 3907905904547764847197154889183844343802986600

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A167010, A309010, A328813, A328814.

a[n_] := Sum[Binomial[n, k]^(n + 1), {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, May 06 2021 *)

{a(n) = sum(k=0, n, binomial(n, k)^(n+1))}

a(n) = A309010(n,n+1) = Sum_{k=0..n} binomial(n,k)^(n+1).

a(n) ~ c * exp(-1/4) * 2^((2*n+1)*(n+1)/2) / (Pi*n)^((n+1)/2), where c = A218792 = Sum_{k = -infinity..infinity} exp(-2*k^2) = 1.271341522189... and c = Sum_{k = -infinity..infinity} exp(-2*(k+1/2)^2) = 1.23528676585389... if n is odd. - Vaclav Kotesovec, May 06 2021

A195907 Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 25 2011

A Riemann sum approximation to Integral_{-oo..oo} exp(-x^2) dx = sqrt(Pi).

			1.77263720482665215303125055115785848134338604537224605383159051...
For comparison, sqrt(Pi) = 1.7724538509055160272981674833411451827975494561223871282138... (A002161).

Mentioned by N. D. Elkies in a lecture on the Poisson summation formula in Nashville TN in May 2010.

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. A002161, A218792.

N[Sum[Exp[-n^2], {n, -Infinity, Infinity}], 200]
RealDigits[ N[ EllipticTheta[3, 0, 1/E], 115]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

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

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

Equals Jacobi theta_{3}(0,exp(-1)). - Jianing Song, Oct 13 2021
Equals eta(i/Pi)^5 / (eta(i/(2*Pi))*eta(2*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
Equals Product_{k>=1} tanh((k*(1 + i*Pi))/2), i=sqrt(-1). - Antonio Graciá Llorente, May 13 2024

A229052 a(n) = Sum_{k=0..n} binomial(n^2-nk, nk-k^2) * binomial(n*k, k^2).

Original entry on oeis.org

1, 2, 6, 92, 6662, 2150552, 3093730764, 18251332286098, 466740831542894470, 47238803741195397513182, 20522607409110459026633535856, 34700017072200465774261952422246668, 250699892545838622857396499800167790109260, 6984916990466628202550631436961441381064765905022

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			The triangle A228832(n,k) = C(n*k, k^2) illustrates the terms involved in the sum a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k):
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; ...

Vincenzo Librandi, Table of n, a(n) for n = 0..50

Cf. A228832, A206847, A218792.

Table[Sum[Binomial[n^2 - n k, n k - k^2] Binomial[n k, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 22 2013 *)

{a(n)=sum(k=0,n,binomial(n^2-n*k,n*k-k^2)*binomial(n*k,k^2))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(n^2-n*k, (n-k)^2) * binomial(n*k, k^2).
a(n) = Sum_{k=0..n} A228832(n, n-k) * A228832(n, k).
a(n) = Sum_{k=0..n} (n^2-n*k)! * (n*k)! / ( ((n-k)^2)! * (n*k-k^2)!^2 * (k^2)! ).
a(n) ~ c * 2^(n^2+2)/(Pi*n^2), where c = EllipticTheta[3,0,1/E^2] = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = 1.23528676585389... if n is odd. - Vaclav Kotesovec, Sep 22 2013

A358495 a(n) = Sum_{k=0..n} binomial(binomial(n, k), n).

Original entry on oeis.org

1, 2, 1, 2, 17, 506, 48772, 13681602, 12287555282, 33669343492094, 311704008906073448, 9309805333008203501246, 987309241535765332024955809, 351345748109942610415182510895442, 459648902729700156671704473390158212154, 2067884865276847662816755891452805155809167114

Offset: 0

Vaclav Kotesovec, Nov 19 2022

nonn

Cf. A357871, A358496.

Table[Sum[Binomial[Binomial[n, k], n], {k, 0, n}], {n, 0, 16}]

a(n) = sum(k=0, n, binomial(binomial(n, k), n)); \\ Michel Marcus, Nov 19 2022

a(n) ~ c * binomial(binomial(n, n/2), n), where c = EllipticTheta[3,0,1/E^2] = JacobiTheta3(0,exp(-2)) = A218792 = 1.271341522189... if n is even and c = EllipticTheta[2,0,1/E^2] = JacobiTheta2(0,exp(-2)) = 1.23528676585389... if n is odd.

Equivalently, a(n) ~ c * 2^(n^2 + n/2 - 1/2) * exp(n - 1/4) / (Pi^((n+1)/2) * n^((3*n+1)/2)).

cp's OEIS Frontend

A167010 a(n) = Sum_{k=0..n} C(n,k)^n.

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A167009 a(n) = Sum_{k=0..n} C(n^2, n*k).

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A328812 Constant term in the expansion of (Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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A195907 Decimal expansion of Sum_{n = -oo..oo} exp(-n^2).

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A229052 a(n) = Sum_{k=0..n} binomial(n^2-n*k, n*k-k^2) * binomial(n*k, k^2).

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A358495 a(n) = Sum_{k=0..n} binomial(binomial(n, k), n).

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A229052 a(n) = Sum_{k=0..n} binomial(n^2-nk, nk-k^2) * binomial(n*k, k^2).