A184731 -id:A184731 - cp's OEIS frontend

A184730 G.f.: exp( Sum_{n>=1} A184731(n)*x^n/n ) where A184731(n) = Sum_{k=0..n} C(n,k)^(k+1).

1, 2, 5, 20, 159, 3152, 168036, 20428850, 5796209814, 4052041564524, 6210335115944263, 21470958882165989854, 183818137919395949397148, 3517964195874870876682733562, 147909303669340763210833833705995, 15391220509661795085065182391703575606

Offset: 0

Paul D. Hanna, Jan 20 2011

nonn

Note that the following g.f. does NOT yield an integer series:

. exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^k] * x^n/n ).

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 159*x^4 + 3152*x^5 +...
log(A(x)) = 2*x + 6*x^2/2 + 38*x^3/3 + 490*x^4/4 + 14152*x^5/5 + 969444*x^6/6 +...+ A184731(n)*x^n/n +...

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

Cf. A184731, A181070, A228899.

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

Equals row sums of triangle A228899.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A184731(k)*a(n-k) for n > 0. - Seiichi Manyama, Jan 10 2019

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

Original entry on oeis.org

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

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

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

A206156 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).

Original entry on oeis.org

1, 2, 6, 92, 5410, 1400652, 2687407464, 18947436116184, 536104663173431874, 130559883231879141946580, 136031455187223511721647272376, 483565526783420050082035900177878504, 14487924180895151383693101563813954330590756

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Ignoring initial term a(0), equals the logarithmic derivative of A206155.

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
where exponentiation yields A206155:
exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
Illustration of initial terms:
a(1) = 1^0 + 1^2 = 2;
a(2) = 1^0 + 2^2 + 1^4 = 6;
a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...

Cf. A206155 (exp), A184731, A206154, A206158, A206152, A220359.

Table[Sum[Binomial[n,k]^(2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

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

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014

A206158 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

Original entry on oeis.org

1, 2, 10, 272, 24226, 12053252, 40086916024, 429254371605824, 23527609330364490754, 10714627376371224032350052, 16964729291782419425708732425300, 109783535843179466164398767001178968704, 6782057095273243388704415924996348722446049600

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Ignoring initial term a(0), equals the logarithmic derivative of A206157.

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 +...
where exponentiation yields A206157:
exp(L(x)) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 + 6686021554*x^6 +...
Illustration of initial terms:
a(1) = 1^1 + 1^3 = 2;
a(2) = 1^1 + 2^3 + 1^5 = 10;
a(3) = 1^1 + 3^3 + 3^5 + 1^7 = 272;
a(4) = 1^1 + 4^3 + 6^5 + 4^7 + 1^9 = 24226;
a(5) = 1^1 + 5^3 + 10^5 + 10^7 + 5^9 + 1^11 = 12053252; ...

Cf. A206157 (exp), A184731, A206154, A206156, A206152, A220359.

Table[Sum[Binomial[n,k]^(2*k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

{a(n)=sum(k=0,n,binomial(n,k)^(2*k+1))}
for(n=0,16,print1(a(n),", "))

Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Mar 03 2014

A219207 Triangle, read by rows, where T(n,k) = binomial(n,k)^(k+1) for n>=0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 27, 1, 1, 16, 216, 256, 1, 1, 25, 1000, 10000, 3125, 1, 1, 36, 3375, 160000, 759375, 46656, 1, 1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1, 1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1, 1, 81, 46656

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 of coefficients C(n,k)^(k+1) begins:
1;
1, 1;
1, 4, 1;
1, 9, 27, 1;
1, 16, 216, 256, 1;
1, 25, 1000, 10000, 3125, 1;
1, 36, 3375, 160000, 759375, 46656, 1;
1, 49, 9261, 1500625, 52521875, 85766121, 823543, 1;
1, 64, 21952, 9834496, 1680700000, 30840979456, 13492928512, 16777216, 1; ...
MATRIX INVERSE.
The matrix inverse starts
1;
-1,1;
3,-4,1;
-73,99,-27,1;
18055,-24496,6696,-256,1;
-55694851,75563975,-20656000,790000,-3125,1; - _R. J. Mathar_, Mar 22 2013

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

Cf. A184731, A219206, A184730.

Table[Binomial[n,k]^(k+1),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2016 *)

{T(n,k)=binomial(n,k)^(k+1)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Row sums equal A184731.

cp's OEIS Frontend

A184730 G.f.: exp( Sum_{n>=1} A184731(n)*x^n/n ) where A184731(n) = Sum_{k=0..n} C(n,k)^(k+1).

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

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

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

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A206156 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).

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A206158 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

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A219207 Triangle, read by rows, where T(n,k) = binomial(n,k)^(k+1) for n>=0, k=0..n.

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