A167010 -id:A167010 - cp's OEIS frontend

A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.

1, 2, 5, 26, 501, 42262, 14564184, 18926665052, 96371663657380, 1825266130738144920, 136764680697906838980633, 38133043109557952095731186822, 42464330390232136488003531922964743

Offset: 0

Paul D. Hanna, Nov 17 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 501*x^4 + 42262*x^5 + ...
log(A(x)) = 2*x + 6*x^2/2 + 56*x^3/3 + 1810*x^4/4 + 206252*x^5/5 + 86874564*x^6/6 + ... + A167010(n)*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..59

Cf. A155200, A167006, A167010.

A167010:= func< n | (&+[Binomial(n,j)^n: j in [0..n]]) >;
function A167007(n)
  if n lt 2 then return n+1;
  else return (&+[A167010(j)*A167007(n-j): j in [1..n]])/n;
  end if; return A167007;
end function;
[A167007(n): n in [0..20]]; // G. C. Greubel, Aug 26 2022

A167010[n_]:= A167010[n]= Sum[Binomial[n,j]^n, {j,0,n}];
A167007[n_]:= A167007[n]= If[n==0, 1, (1/n)*Sum[A167010[j]*A167007[n-j], {j,n}]];
Table[A167007[n], {n,0,30}] (* G. C. Greubel, Aug 26 2022 *)

{a(n) = polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m,k)^m)*x^m/m) +x*O(x^n)), n)};

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sum(j=0, k, binomial(k, j)^k)*a(n-k)))} \\ Paul D. Hanna, Nov 25 2009

def A167010(n): return sum(binomial(n,j)^n for j in (0..n))
def A167007(n): return 1 if (n==0) else (1/n)*sum( A167010(j)*A167007(n-j) for j in (1..n))
[A167007(n) for n in (0..30)] # G. C. Greubel, Aug 26 2022

a(n) = (1/n)*Sum_{k=1..n} A167010(k)*a(n-k) for n>0 with a(0)=1. - Paul D. Hanna, Nov 25 2009

A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

Original entry on oeis.org

1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584

Offset: 0

Paul D. Hanna, Nov 17 2009

Row sums of A219206.

Reinhard Zumkeller, Table of n, a(n) for n = 0..75
Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.

a167008 = sum . a219206_row  -- Reinhard Zumkeller, Feb 27 2015

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

Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,20}]}]
(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2},WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)
Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* Harvey P. Dale, Oct 19 2021 *)

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

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

Limit_{n->oo} a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012

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

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11

Offset: 0

Seiichi Manyama, Jul 06 2019

A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019
Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below).  Then A(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

			Square array, A(n, k), begins:
   1,  1,   1,    1,     1,      1, ... A000012;
   2,  2,   2,    2,     2,      2, ... A007395;
   3,  4,   6,   10,    18,     34, ... A052548;
   4,  8,  20,   56,   164,    488, ... A115099;
   5, 16,  70,  346,  1810,   9826, ...
   6, 32, 252, 2252, 21252, 206252, ...
Antidiagonals, T(n, k), begin:
  1;
  1,  2;
  1,  2,   3;
  1,  2,   4,    4;
  1,  2,   6,    8,    5;
  1,  2,  10,   20,   16,     6;
  1,  2,  18,   56,   70,    32,     7;
  1,  2,  34,  164,  346,   252,    64,    8;
  1,  2,  66,  488, 1810,  2252,   924,  128,   9;
  1,  2, 130, 1460, 9826, 21252, 15184, 3432, 256,  10;

R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Main diagonal gives A167010.
Cf. A328747, A328748, A328807, A328812.
Cf. A000012, A007395, A052548, A115099.

[(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022

nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)

A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022

flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 26 2022

A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
A(n, n+1) = A328812(n).
A(n, n) = A167010(n).
T(n, k) = A(k, n-k) (antidiagonals).
T(n, n) = A000027(n+1).
T(n, n-1) = A000079(n-1).
T(n, n-2) = A000984(n-2).
T(n, n-3) = A000172(n-3).
T(n, n-4) = A005260(n-4).
T(n, n-5) = A005261(n-5).
T(n, n-6) = A069865(n-6).
T(n, n-7) = A182421(n-7).
T(n, n-8) = A182422(n-8).
T(n, n-9) = A182446(n-9).
T(n, n-10) = A182447(n-10).
T(n, n-11) = A342294(n-11).
T(n, n-12) = A342295(n-12).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

Original entry on oeis.org

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465

Offset: 1

Amarnath Murthy, Jul 04 2004

nonn

The product of the terms of the n-th row is given by A034841.

Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..338

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018

A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007

Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)

a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

More terms from R. J. Mathar, Apr 30 2007

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

Original entry on oeis.org

1, 2, 9, 103, 2073, 58481, 2101813, 91492906, 4671050401, 273437232283, 18046800575211, 1325445408799007, 107200425419863009, 9466283137384124247, 906151826270369213655, 93459630239922214535911, 10331984296666203358431361, 1218745075041575200343722415

Offset: 0

Vaclav Kotesovec, Jun 06 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A072034, A086331, A096131, A167008, A167010, A349471.

[(&+[Binomial(n*j,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 31 2022

Table[Sum[Binomial[k*n, k], {k, 0, n}], {n, 0, 20}]

A226391(n):=sum(binomial(k*n,k), k,0,n); makelist(A226391(n),n,0,30); /* Martin Ettl, Jun 06 2013 */

@CachedFunction
def A226391(n): return sum(binomial(n*j, j) for j in (0..n))
[A226391(n) for n in (0..30)] # G. C. Greubel, Aug 31 2022

a(n) ~ binomial(n^2, n).

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

A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

Original entry on oeis.org

1, 2, 13, 352, 38401, 16971876, 29359436149, 207003074670848, 5679112509686022145, 636468045901197095750500, 277939985126193076692203962501, 494649880078824954885176565423811200, 3447375085398645453825889951638344722092289, 97424105704407389799712313421357308088296084669504

Offset: 0

Seiichi Manyama, Jul 11 2020

Seiichi Manyama, Table of n, a(n) for n = 0..57
Vaclav Kotesovec, Plot of a(n) / (2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2)) for n = 1..10000000

Main diagonal of A336187.

Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214.

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

Unprotect[Power]; 0^0 = 1; a[n_] := Sum[n^k * Binomial[n, k]^n, {k, 0, n} ]; Array[a, 14, 0] (* Amiram Eldar, Jul 11 2020 *)

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

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

Let f(n) = 2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2). For sufficiently large n 0.7675... < a(n)/f(n) < 0.7900... - Vaclav Kotesovec, Jul 11 2020
The above bounds of Vaclav Kotesovec can be recast as: |a(n)/f(n) - exp(-1/4)| <= (3*Pi)^(-2) for sufficiently large n. - Peter Luschny, Jul 12 2020
a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-2), exp(-4)) * 2^(n^2 + n/2) / Pi^(n/2) if n is even and a(n) ~ exp(-3/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-4), exp(-4)) * 2^(n^2 + n/2) * sqrt(n)  / Pi^(n/2) if n is odd. - Vaclav Kotesovec, Jul 13 2020

A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.

Original entry on oeis.org

1, 3, 13, 171, 7761, 1256283, 741398869, 1609036666443, 13118066779885825, 399221556627301207443, 46476897754761801245056293, 20377119057713827002258336842283, 34592895120825704155462768381947657489, 222457046333769635263635086646525921070978443

Offset: 0

Seiichi Manyama, Jul 11 2020

nonn

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

Main diagonal of A336203.

Cf. A167010, A336188, A336202, A336212.

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

Table[Sum[2^k*Binomial[n, k]^n, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 12 2020 *)

{a(n) = sum(k=0, n, 2^k*binomial(n, k)^n)};

def A336204(n): return sum(2^k*binomial(n,k)^n for k in (0..n))
[A336204(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) ~ c * 2^(n*(n+1)) / (Pi*n)^(n/2), where c = exp(-1/4) * Sum_{k = -oo..oo} 2^k * exp(-2*k^2) = 1.0434092897163574491113380912895917... if n is even and c = exp(-1/4) * Sum_{k = -oo..oo} 2^(k + 1/2) * exp(-2*(k + 1/2)^2) = 1.029587234777114329090639723058125257... if n is odd. - Vaclav Kotesovec, Jul 12 2020

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

cp's OEIS Frontend

A167007 G.f.: A(x) = exp( Sum_{n>=1} A167010(n)*x^n/n ) where A167010(n) = Sum_{k=0..n} binomial(n,k)^n.

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A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

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

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A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

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A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

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A226391 a(n) = Sum_{k=0..n} binomial(k*n, k).

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