A167010 -id:A167010 - 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

A209427 T(n,k) = binomial(n,k)^n.

Original entry on oeis.org

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, 1, 823543, 1801088541, 64339296875, 64339296875, 1801088541, 823543, 1, 1, 16777216, 377801998336, 96717311574016, 576480100000000, 96717311574016, 377801998336, 16777216, 1

Offset: 0

Paul D. Hanna, Mar 08 2012

Row sums equals A167010.
Column 1 forms A000312.
Antidiagonal sums form A209428.

			This triangle 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;
1, 823543, 1801088541, 64339296875, 64339296875, 1801088541, 823543, 1;
1, 16777216, 377801998336, 96717311574016, 576480100000000, 96717311574016, 377801998336, 16777216, 1; ...

Paul D. Hanna, Rows n = 0..30, flattened.

Cf. A167010 (row sums), A000312 (column 1), A209428.

Table[Binomial[n,k]^n, {n,0,10}, {k,0,n}]// Flatten (* G. C. Greubel, Jan 03 2018 *)

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

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.

A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

Original entry on oeis.org

1, 3, 15, 381, 67635, 83118753, 813824623689, 58040410068847251, 32150480245981639533315, 154935057570894645075940703673, 5474671509704049919709361235659936825, 1600436120524545216094358662984789029130593831

Offset: 0

Ilya Gutkovskiy, Jul 15 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..47

Cf. A000244, A002893, A141057, A167010, A172434, A180350.

Table[Sum[Sum[(Binomial[n, k] Binomial[k, j])^n, {j, 0, k}], {k, 0, n}], {n, 0, 11}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^3, {x, 0, n}], {n, 0, 11}]

a(n) = sum(k=0, n, sum(j=0, k, (binomial(n,k) * binomial(k,j))^n)); \\ Michel Marcus, Jul 16 2020

a(n) = (n!)^n * [x^n] (Sum_{k>=0} x^k / (k!)^n)^3.

A328810 a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

Original entry on oeis.org

1, 3, 11, 93, 2583, 260613, 99915029, 144072750195, 808177412109895, 16892305881120020613, 1388286655114683125139201, 423109739462061163278604529475, 511885816860737850466697173188711669, 2296554708428991868313593456071099604464483

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Main diagonal of A328807.

Cf. A167010, A328811.

Table[Sum[Binomial[n, i]*Sum[Binomial[i, j]^n, {j, 0, i}], {i, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, binomial(n, i)*sum(j=0, i, binomial(i, j)^n))}

a(n) ~ A167010(n). - Vaclav Kotesovec, Oct 28 2019

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^n.

Original entry on oeis.org

1, 1, 3, 31, 1255, 161671, 75481581, 121338954577, 734884394666535, 15970479086714049751, 1347242827078365957146473, 415839472158527880691583531617, 507266883682599825619985300960971525, 2284735689605775548174387143718048664963601

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Main diagonal of A328747.

Cf. A167010, A328810.

Table[Sum[(-1)^(n-i)*Binomial[n, i]*Sum[Binomial[i, j]^n, {j, 0, i}], {i, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^n))}

a(n) ~ A167010(n). - Vaclav Kotesovec, Oct 28 2019

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.

A295611 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^k.

Original entry on oeis.org

1, 0, 0, 6, -30, -280, 35070, -2508268, -47103462, 241470400824, -256752145545390, 128291714550379292, 2203924344437376054780, -37693423679943326954848176, 485163732930867224220253809178, 27101025121379607823580070619517816

Offset: 0

Ilya Gutkovskiy, Nov 24 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..75
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A167008, A167010, A219206.

Table[Sum[(-1)^k Binomial[n, k]^k, {k, 0, n}], {n, 0, 15}]
Table[Sum[(-1)^k (n!/(k! (n - k)!))^k, {k, 0, n}], {n, 0, 15}]

a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^k); \\ Michel Marcus, Nov 25 2017

a(n) = Sum_{k=0..n} (-1)^k*A219206(n,k).

Limit n->infinity |a(n)|^(1/n^2) = r^(r^2/(1-2*r)) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243096929661621777... is the real root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Nov 25 2017

A336829 a(n) = Sum_{k=0..n} binomial(n+k,k)^n.

Original entry on oeis.org

1, 3, 46, 9065, 25561876, 1048567813062, 632156164654144530, 5652307059542612442465921, 755658094192422806457805924637704, 1521188219372604726826961340683399629967888, 46388428590466766659538640978460161019178279424832676

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..41

Cf. A001700, A112028, A112029, A167010, A219562, A219563, A219564.

[(&+[Binomial(2*n-j,n)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Binomial[n + k, k]^n, {k, 0, n}], {n, 0, 10}]

a(n) = sum(k=0, n, binomial(n+k, k)^n); \\ Michel Marcus, Aug 05 2020

def A336829(n): return sum(binomial(2*n-j, n)^n for j in (0..n))
[A336829(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi*n)^(n/2). - Vaclav Kotesovec, Jul 10 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A209427 T(n,k) = binomial(n,k)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A336270 a(n) = Sum_{k=0..n} Sum_{j=0..k} (binomial(n,k) * binomial(k,j))^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328810 a(n) = Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^n.