A167008 -id:A167008 - cp's OEIS frontend

A220359 Decimal expansion of the root of the equation (1-r)^(2r-1) = r^(2r).

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

Offset: 0

Vaclav Kotesovec, Dec 12 2012

Constant is associated with A167008, A219206 and A219207.

			0.70350607643066243...

Cf. A167008, A219206, A219207, A228808, A184731, A206154, A206156, A206158, A237421, A295611.

Digits:= 140:
v:= convert(fsolve( (1-r)^(2*r-1) = r^(2*r), r=1/2), string):
seq(parse(v[n+2]), n=0..120);  # Alois P. Heinz, Dec 12 2012

RealDigits[r/.FindRoot[(1-r)^(2*r-1)==r^(2*r),{r,1/2}, WorkingPrecision->250], 10, 200][[1]]

solve(x=.7,1,(1-x)^(2*x-1) - x^(2*x)) \\ Charles R Greathouse IV, Apr 25 2016

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).

A184731 a(n) = Sum_{k=0..n} C(n,k)^(k+1).

Original entry on oeis.org

1, 2, 6, 38, 490, 14152, 969444, 140621476, 46041241698, 36363843928316, 62022250535177416, 236043875222171125276, 2205302277098968939256248, 45728754995013679582534494332, 2070631745797418828103776968679204

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

			The terms begin:
a(0) = 1;
a(1) = 1 + 1^2 = 2;
a(2) = 1 + 2^2 + 1^3 = 6;
a(3) = 1 + 3^2 + 3^3 + 1^4 = 38;
a(4) = 1 + 4^2 + 6^3 + 4^4 + 1^5 = 490;
a(5) = 1 + 5^2 + 10^3 + 10^4 + 5^5 + 1^6 = 14152.

Seiichi Manyama, Table of n, a(n) for n = 0..74

Cf. A184730, A167008, A220359.

Table[Sum[Binomial[n, k]^(k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 29 2014 *)

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

Forms the logarithmic derivative of A184730 (ignoring the initial term).

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.53362806511..., where r = 0.70350607643... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Jan 29 2014

A219206 Triangle, read by rows, where T(n,k) = binomial(n,k)^k for n>=0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 9, 1, 1, 4, 36, 64, 1, 1, 5, 100, 1000, 625, 1, 1, 6, 225, 8000, 50625, 7776, 1, 1, 7, 441, 42875, 1500625, 4084101, 117649, 1, 1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152, 1, 1, 9, 1296, 592704, 252047376, 31757969376, 351298031616, 78364164096, 43046721, 1

Offset: 0

Paul D. Hanna, Nov 14 2012

Maximal term in row n is asymptotically in position k = r*n, where r = A220359 = 0.70350607643... is a root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Nov 15 2012

			Triangle begins:
  1;
  1, 1;
  1, 2,   1;
  1, 3,   9,      1;
  1, 4,  36,     64,        1;
  1, 5, 100,   1000,      625,         1;
  1, 6, 225,   8000,    50625,      7776,         1;
  1, 7, 441,  42875,  1500625,   4084101,    117649,       1;
  1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152,  1;
  ...

Paul D. Hanna, Rows n = 0..45, flattened.

Cf. A167008 (row sums).

Cf. A002262, A007318, A219207.

a219206 n k = a219206_tabl !! n !! k
a219206_row n = a219206_tabl !! n
a219206_tabl = zipWith (zipWith (^)) a007318_tabl a002262_tabl
-- Reinhard Zumkeller, Feb 27 2015

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

Row sums equal A167008.

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

Original entry on oeis.org

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

A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

Original entry on oeis.org

1, 1, 2, 27, 9216, 312500000, 4251528000000000, 95432797246104853383515625, 14719075154533285649961930052505436160000, 65577306173662530591576256095315195684570038194755952705536

Offset: 0

Vaclav Kotesovec, Apr 14 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..22

Cf. A001142, A067055, A167008, A255268, A272093.

Table[Product[Binomial[n, k]^k, {k, 0, n}], {n, 0, 10}]
Table[(n!)^(n*(n+1)/2) / BarnesG[n+2]^n, {n, 0, 10}]

a(n) = prod(k=0, n, binomial(n,k)^k); \\ Michel Marcus, Apr 14 2023

a(n) = Product_{k=0..n} n!^k / k!^n.
a(n) = A067055(n) / A255268(n).
a(n) ~ A^n * exp((6*n^3 + 12*n^2 - n - 1)/24) / ((2*Pi)^(n*(n+1)/4) * n^(n*(3*n+2)/12)), where A is the Glaisher-Kinkelin constant A074962.

A206154 a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).

Original entry on oeis.org

1, 2, 10, 110, 2386, 125752, 14921404, 3697835668, 2223231412546, 3088517564289836, 9040739066816429380, 63462297965044771663708, 1064766030857977088480630740, 37863276208844960432962611293828, 3144384748384240804260912067907833280

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Ignoring initial term a(0), equals the logarithmic derivative of A206153.

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
Illustration of initial terms:
a(1) = 1^2 + 1^3 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...

Cf. A206153 (exp), A184731, A206156, A206158, A206152, A167008, A220359.

Table[Sum[Binomial[n,k]^(k+2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 16 2014 *)

{a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
for(n=0,16,print1(a(n),", "))

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.53362806511..., where r = 0.70350607643... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Jan 29 2014

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

cp's OEIS Frontend

A220359 Decimal expansion of the root of the equation (1-r)^(2*r-1) = r^(2*r).

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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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A184731 a(n) = Sum_{k=0..n} C(n,k)^(k+1).

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A219206 Triangle, read by rows, where T(n,k) = binomial(n,k)^k for n>=0, k=0..n.

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

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A362288 a(n) = Product_{k=0..n} binomial(n,k)^k.

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A220359 Decimal expansion of the root of the equation (1-r)^(2r-1) = r^(2r).