A323778 -id:A323778 - cp's OEIS frontend

A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)x^k = Sum_{n>=1} a(n)x^n/n.

1, 3, 7, 15, 231, 2763, 37773, 3347359, 145164760, 15115517783, 5300285945494, 841490209145991, 700215432847179640, 821522962294608211319, 580955012898669141073842, 3240132942509582109732641935, 12114306457535986210506222037102

Offset: 1

Paul D. Hanna, Feb 15 2012

nonn

Equals the logarithmic derivative of A206850.

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 231*x^5/5 + 2763*x^6/6 +...
where exponentiation yields the g.f. of A206850:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
By definition, the l.g.f. equals the series:
L(x) = (C(1,0) + C(1,1)*x)*x
+ (C(4,0) + C(4,1)*x + C(4,4)*x^2)*x^2/2
+ (C(9,0) + C(9,1)*x + C(9,4)*x^2 + C(9,9)*x^3)*x^3/3
+ (C(16,0) + C(16,1)*x + C(16,4)*x^2 + C(16,9)*x^3 + C(16,16)*x^4)*x^4/4
+ (C(25,0) + C(25,1)*x + C(25,4)*x^2 + C(25,9)*x^3 + C(25,16)*x^4 + C(25,25)*x^5)*x^5/5 +...
More explicitly,
L(x) = (1 + 1*x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 9*x + 126*x^2 + 1*x^3)*x^3/3
+ (1 + 16*x + 1820*x^2 + 11440*x^3 + 1*x^4)*x^4/4
+ (1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + 1*x^5)*x^5/5
+ (1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + 1*x^6)*x^6/6 +...

Seiichi Manyama, Table of n, a(n) for n = 1..94

Cf. A206850 (exp), A206847, A206849, A123165.

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

{a(n)=n*polcoeff(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k^2)*x^k)+x*O(x^n)),n)}

{a(n)=n*sum(k=0,n\2, binomial((n-k)^2, k^2)/(n-k))}
for(n=1, 20, print1(a(n), ", "))

a(n) = n * Sum_{k=0..[n/2]} binomial((n-k)^2, k^2)/(n-k).

Limit n->infinity a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166349..., where r = A323778 = 0.365498498219858044579... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 03 2014

A123165 Row sums of A123163.

Original entry on oeis.org

1, 1, 2, 5, 11, 143, 1847, 24127, 2101931, 96398196, 9362963203, 3376252046640, 551993132054154, 434634824535802596, 528116646162507517308, 372831439174848001477184, 2029862948426766042724907818

Offset: 0

Roger L. Bagula, Oct 02 2006

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..94

Cf. A123163, A206851.

[(&+[Binomial((n-k)^2, k^2): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jul 19 2023

A123163[n_, k_]= ((n-k)^2)!/((k^2)!(n^2-2*n*k)!);
Table[Sum[A123163[n,k], {k,0,n/2}], {n,0,20}]
Table[Sum[Binomial[(n-k)^2,k^2], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 04 2014 *)

{a(n) = sum(k=0, n\2, binomial((n-k)^2, k^2))} \\ Seiichi Manyama, Jan 28 2019

def A123165(n): return sum(binomial((n-k)^2, k^2) for k in range(n//2+1))
[A123165(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

Limit_{n-> oo} a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166..., where r = A323778 = 0.365498498219858044579736... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 04 2014

Edited by N. J. A. Sloane, Oct 04 2006

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.2206760413237406963128222699980501671876810310275740395417335127215630565...

Cf. A166895, A209331, A209428.

Cf. A220359, A237421, A245259, A323768, A323773, A323778.

RealDigits[r/.FindRoot[(1-2*r)^(3-4*r) == (1-r)^(2-2*r) * r^(1-2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

A323773 Decimal expansion of the root of the equation (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 27 2019

			0.3663201503052830964087236563781171194011826607210994595491823160184...

Cf. A220359, A228833, A237421, A245259, A323768, A323777, A323778.

RealDigits[r/.FindRoot[(1-2*r)^(4*r-1) * (1-r)^(1-2*r) == r^(2*r), {r, 1/3}, WorkingPrecision->250], 10, 200][[1]]

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A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n.

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A123165 Row sums of A123163.

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A323777 Decimal expansion of the root of the equation (1-2*r)^(3-4*r) = (1-r)^(2-2*r) * r^(1-2*r).

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A323773 Decimal expansion of the root of the equation (1-2*r)^(4*r-1) * (1-r)^(1-2*r) = r^(2*r).

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A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)x^k = Sum_{n>=1} a(n)x^n/n.

A323777 Decimal expansion of the root of the equation (1-2r)^(3-4r) = (1-r)^(2-2r) r^(1-2*r).

A323773 Decimal expansion of the root of the equation (1-2r)^(4r-1) * (1-r)^(1-2r) = r^(2r).