A336204 -id:A336204 - cp's OEIS frontend

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..56

Cf. A072034, A086331, A167008, A167010, A328812, A336188, A336204, A336212, A336213.

Flatten[{1, Table[Sum[k^n*Binomial[n, k]^n, {k, 1, n}], {n, 1, 15}]}]

a(n) = if (n==0, 1, sum(k=0, n, k^n * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.

a(n) = n^n * A328812(n-1) for n > 0. - Seiichi Manyama, Jul 15 2020

A336202 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^n.

Original entry on oeis.org

1, 0, -3, 136, 3585, -8065624, 985282165, 102324513620736, -758462117693095935, -310124007268556369914448, 59000420766060452235999162501, 231739512209034254162941881236647760, -948238573709799908746228205852168505192191, -43263440520748047736633474769642007589423961473152

Offset: 0

Seiichi Manyama, Jul 11 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..57

Main diagonal of A336201.

Cf. A167010, A336188, A336204.

a[n_] := Sum[If[n == k == 0, 1, (-n)^k] * Binomial[n, k]^n, {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, May 01 2021 *)

{a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^n)}

A336212 a(n) = Sum_{k=0..n} 3^k * binomial(n,k)^n.

Original entry on oeis.org

1, 4, 22, 352, 19426, 3862744, 2764634356, 7403121210496, 73087416841865890, 2751096296949421766824, 387442256655054793494004132, 210421903024207931092658380560256, 431805731803048897945138363105712865124, 3443300668674111298036287560913860498279204224

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

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

Cf. A167010, A336188, A336204.

Table[Sum[3^k * Binomial[n, k]^n, {k, 0, n}], {n, 0, 15}]

{a(n) = sum(k=0, n, 3^k*binomial(n, k)^n)} \\ Seiichi Manyama, Jul 13 2020

a(n) ~ c * exp(-1/4) * 2^(n^2 + n/2) * (3/(Pi*n))^(n/2), where c = Sum_{k = -infinity..infinity} 3^k * exp(-2*k^2) = 1.4541744598397064657680975624481... if n is even and c = Sum_{k = -infinity..infinity} 3^(k + 1/2) * exp(-2*(k + 1/2)^2) = 1.4606428581939532945566671970305... if n is odd.

A336213 a(n) = Sum_{k=0..n} k^k * binomial(n,k)^n, with a(0)=1.

Original entry on oeis.org

1, 2, 9, 163, 12609, 3906251, 4835455813, 23882051929709, 470073929716006913, 36867039626275056203923, 11562789460238169439667262501, 14393917436542502296957220221339601, 72060131612303615870363237649174605005057, 1424448870088911493303605765206905153730451241313

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Plot of a(n)/f(n) for n = 1..10000000

Cf. A072034, A086331, A167008, A167010, A336188, A336204, A336212, A336214.

Table[1 + Sum[k^k * Binomial[n, k]^n, {k, 1, n}], {n, 0, 15}]

a(n) = if (n==0, 1, sum(k=0, n, k^k * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

Let f(n) = exp(-1/4) * QPochhammer(exp(-4)) * 2^(n^2 - 1/4) * exp((3*log(n)^2 + 3*log(2)^2 + Pi^2 - 1)/24) * n^((1 - log(2))/4) / Pi^(n/2). For sufficiently large n 0.985... < a(n)/f(n) < 1.015...

a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-1)/2, exp(-4)) * 2^(n^2) / Pi^(n/2) if n is even and a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-3)/2, exp(-4)) * sqrt(n) * 2^(n^2 - 1/2) / Pi^(n/2) if n is odd.

A336203 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j)^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 7, 1, 3, 9, 15, 1, 3, 13, 27, 31, 1, 3, 21, 63, 81, 63, 1, 3, 37, 171, 321, 243, 127, 1, 3, 69, 495, 1521, 1683, 729, 255, 1, 3, 133, 1467, 7761, 14283, 8989, 2187, 511, 1, 3, 261, 4383, 41361, 131283, 138909, 48639, 6561, 1023, 1, 3, 517, 13131, 227601, 1256283, 2336629, 1385163, 265729, 19683, 2047

Offset: 0

Seiichi Manyama, Jul 11 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - 2 * Product_{j=1..k} x_j) for k>0.

			Square array begins:
   1,   1,    1,     1,      1,       1, ...
   3,   3,    3,     3,      3,       3, ...
   7,   9,   13,    21,     37,      69, ...
  15,  27,   63,   171,    495,    1467, ...
  31,  81,  321,  1521,   7761,   41361, ...
  63, 243, 1683, 14283, 131283, 1256283, ...

Columns k=0-4 give: A000225(n+1), A000244, A001850, A206178, A216696.
Main diagonal gives A336204.
Cf. A309010.

T[n_, k_] := Sum[2^j * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

cp's OEIS Frontend

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

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A336202 a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^n.

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A336212 a(n) = Sum_{k=0..n} 3^k * binomial(n,k)^n.

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A336213 a(n) = Sum_{k=0..n} k^k * binomial(n,k)^n, with a(0)=1.

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A336203 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j)^k.

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