A177385 -id:A177385 - cp's OEIS frontend

A249748 Decimal expansion of a constant related to A177385.

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

Offset: 1

Vaclav Kotesovec, Nov 04 2014

			1.0468991926259542411134251832531181797678904647564718411558474458277757...

Cf. A177385, A177386.

Equals limit n->infinity (A177385(n)/(n!)^2)^(1/n).

Equals limit n->infinity (A177386(n)/n!)^(1/n) / 2. - Vaclav Kotesovec, Nov 06 2014

A193467 E.g.f.: Sum_{n>=0} x^n * exp(n(n+1)/2x).

Original entry on oeis.org

1, 1, 4, 27, 280, 4025, 75876, 1800253, 52193408, 1807302897, 73406128420, 3446236588421, 184750419871920, 11194423784630281, 759960096829452260, 57367378069894391325, 4783586470578255085696, 438054092182322814028001, 43827052650093379145736900

Offset: 0

Paul D. Hanna, Jul 27 2011

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 280*x^4/4! + 4025*x^5/5! + 75876*x^6/6! + 1800253*x^7/7! +...
where
A(x) = 1 + x*exp(x) + x^2*exp(3*x) + x^3*exp(6*x) + x^4*exp(10*x) +...
By a q-series identity:
A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x*exp(2*x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x*exp(3*x))/((1-x*exp(2*x))*(1-x*exp(4*x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x*exp(3*x))*(1-x*exp(5*x))/((1-x*exp(2*x))*(1-x*exp(4*x))*(1-x*exp(6*x))) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A193421, A193469, A177385, A193466.

{a(n)=local(Egf); Egf=sum(m=0, n, x^m*exp(m*(m+1)/2*x+x*O(x^n))); n!*polcoeff(Egf, n)}

/* q-series identity: */
{a(n)=local(A=1+x);for(i=1, n, A=sum(m=0, n, x^m*exp(m*x+x*O(x^n))*prod(k=1, m, (1-x*exp((2*k-1)*x+x*O(x^n)))/(1-x*exp((2*k)*x+x*O(x^n)))))); n!*polcoeff(A, n)}

E.g.f.: A(x) = Sum_{n>=0} x^n*exp(n*x)*Product_{k=1..n} (1 - x*exp((2*k-1)*x)) / (1 - x*exp(2*k*x)), due to a q-series identity.
Let q = exp(x), then the e.g.f. equals the continued fraction:
A(x) = 1/(1- q*x/(1- q*(q-1)*x/(1- q^3*x/(1- q^2*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^3-1)*x/(1- q^7*x/(1- q^4*(q^4-1)*x/(1- ...))))))))), due to a partial theta function identity.
O.g.f.: Sum_{k>=0} k! * x^k / (1 - binomial(k+1,2)*x)^(k+1). - Ilya Gutkovskiy, Jul 02 2019

A177387 E.g.f.: Sum_{n>=0} Product_{k=1..n} sin(k*x).

Original entry on oeis.org

1, 1, 4, 35, 536, 12721, 432364, 19923455, 1195597616, 90597432961, 8459910749524, 954441965659775, 127987398340965896, 20120987017230590401, 3665273670382984503484, 765857737574513717138495

Offset: 0

Paul D. Hanna, May 15 2010

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 35*x^3/3! + 536*x^4/4! +...
A(x) = 1 + sin(x) + sin(x)*sin(2x) + sin(x)*sin(2x)*sin(3x) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..253
Vaclav Kotesovec, Graph of 1000 terms (limit to constant c)

Cf. A177388, A177385.

Flatten[{1,Rest[CoefficientList[Series[Sum[Product[Sin[m*x],{m,1,k}],{k,1,20}],{x,0,20}],x] * Range[0,20]!]}] (* Vaclav Kotesovec, Nov 03 2014 *)
nn=20; tab=ConstantArray[0,nn]; tab[[1]]=Series[Sin[x],{x,0,nn}]; Do[tab[[k]]=Series[tab[[k-1]]*Sin[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Rest[CoefficientList[Sum[tab[[k]],{k,1,nn}],x]*Range[0,nn]!]}] (* Vaclav Kotesovec, Nov 03 2014 (more efficient) *)

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sin(k*X)));n!*polcoeff(Egf,n)}

a(n) ~ c * 2^(n+1) * n^(2*n+7/6) / (Pi^(n-1) * exp(2*n) * (log(2))^n), where c = 1.01529686... . - Vaclav Kotesovec, Nov 03 2014

A177386 O.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*arcsinh(2x)).

Original entry on oeis.org

1, 2, 8, 48, 400, 4192, 52720, 773536, 12970016, 244625088, 5125896112, 118137655840, 2970016739552, 80883641686848, 2372035401856352, 74528583049288768, 2497667361588205632, 88932255196677684608

Offset: 0

Paul D. Hanna, May 15 2010

nonn

Lim_{n->infinity} n!*A177386(n) / (2^n*A177385(n)) = 1. - Vaclav Kotesovec, Nov 06 2014

			O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 400*x^4 + 4192*x^5 + ...
Let G(x) be the e.g.f. of A177385:
G(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 616*x^4/4! + 16081*x^5/5! + ...
then A(x) = G(arcsinh(2x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A177385, A249748.

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sinh(k*asinh(2*X))));polcoeff(Egf,n)}

O.g.f.: A(x) = G(arcsinh(2x)) where G(x) = e.g.f. of A177385.

a(n) ~ c * d^n * n!, where d = 2*A249748 = 2.0937983852519084822268503..., c = 0.880333778211172907563073... (constant c is same as for A177385). - Vaclav Kotesovec, Nov 06 2014

A193466 E.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} cosh(k*x).

Original entry on oeis.org

1, 1, 2, 9, 84, 965, 12750, 225967, 5241880, 139776345, 4272148890, 155402034491, 6513558987540, 304210965928597, 15965624278036342, 941149313037711975, 61160783460181817520, 4356686998946564113457, 340627068039399668576946, 29015657457166019702796787

Offset: 0

Paul D. Hanna, Jul 27 2011

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 84*x^4/4! + 965*x^5/5! + 12750*x^6/6! + 225967*x^7/7! +...
where
A(x) = 1 + x*cosh(x) + x^2*cosh(x)*cosh(2*x) + x^3*cosh(x)*cosh(2*x)*cosh(3*x) + x^4*cosh(x)*cosh(2*x)*cosh(3*x)*cosh(4*x) +...
Also,
A(x) = 1 + x*exp(-x)*(1+exp(2*x))/2 + x^2*exp(-3*x)*(1+exp(2*x))*(1+exp(4*x))/2^2 +  x^3*exp(-6*x)*(1+exp(2*x))*(1+exp(4*x))*(1+exp(6*x))/2^3 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A177385, A193467, A193468.

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, x^m*prod(k=1, m, cosh(k*X))); n!*polcoeff(Egf, n)}

E.g.f.: Sum_{n>=0} (x/2)^n * exp(-n*(n+1)*x/2) * Product_{k=1..n} (1 + exp(2*k*x)).

A249564 a(n) = Sum_{k = 0..n} (k*(k+1)/2)^n.

Original entry on oeis.org

1, 1, 10, 244, 11378, 867395, 98204132, 15475158552, 3239399341956, 869652788703285, 291315412833808702, 119114020598815073524, 58386684085633233147478, 33797341113242898165287495, 22810507257314647778044971848, 17755122836243141585656207243952

Offset: 0

Vaclav Kotesovec, Nov 01 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000217, A031971, A177385.

Table[n!*SeriesCoefficient[Sum[Exp[x*k*(k+1)/2], {k, 0, n}], {x, 0, n}], {n, 0, 20}]
Flatten[{1,Table[Sum[(k*(k+1)/2)^n,{k,1,n}],{n,1,20}]}]

a(n) = sum(k=0, n, (k*(k+1)/2)^n); \\ Michel Marcus, Aug 24 2023

E.g.f.: Sum_{n>=0} exp(x*n*(n+1)/2).

a(n) ~ exp(3) * n^(2*n) / ((exp(2)-1) * 2^n).

New name from Peter Bala, Aug 18 2023

A249698 E.g.f.: Sum_{n>=0} Product_{k=1..n} cosh(k*x).

Original entry on oeis.org

1, 0, 6, 0, 2426, 0, 7553776, 0, 90192976308, 0, 2939813898295990, 0, 213701821328573755046, 0, 30292525174041077292043440, 0, 7609302838629919155170452856136, 0, 3152886110080180503361685427596189430, 0, 2038143533263759863560759054752335955960482

Offset: 0

Vaclav Kotesovec, Nov 04 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..234

Cf. A177385, A224899, A249489, A177387, A249737.

Table[n!*SeriesCoefficient[Sum[Product[Cosh[k*x],{k,1,j}],{j,0,n}],{x,0,n}],{n,0,20}]
nn=20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Cosh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Cosh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Table[kk!*Sum[Coefficient[tab[[k]],x^kk],{k,1,kk}],{kk,1,nn}]}] (* more efficient *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, prod(k=1, m, cosh(k*X))); n!*polcoeff(Egf, n)}
for(n=0, 20, print1(a(n), ", "))

If n is even a(n) ~ c * d^n * n^(2*n) / (2^(2*n-2) * exp(2*n)), where d = 8.9061971328050809899679389417314..., c = 1.243878632396819914960247452516...

A249737 E.g.f.: Product_{k=1..n} cosh(k*x).

Original entry on oeis.org

1, 0, 5, 0, 1992, 0, 6167551, 0, 73432708224, 0, 2389444075877425, 0, 173496878823412858880, 0, 24573448663070711791073155, 0, 6168942712247503719875933929472, 0, 2554865971518520622455831203134760669, 0, 1650933998542152349112398040415912949710848, 0

Offset: 0

Vaclav Kotesovec, Nov 04 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..235

Cf. A249698, A177385, A177387, A001044 (e.g.f.: product_{k=1..n} sinh(k*x)).

Table[n!*SeriesCoefficient[Product[Cosh[k*x],{k,1,n}],{x,0,n}],{n,0,20}]
nn = 20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Cosh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Cosh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Table[k!*Coefficient[tab[[k]],x^k],{k,1,nn}]}] (* more efficient *)

If n is even a(n) ~ c * d^n * n^(2*n) / (2^(2*n-2) * exp(2*n)), where d = 8.9061971328050809899679389417314..., c = 1.004120096780056350248342856...

cp's OEIS Frontend

A249748 Decimal expansion of a constant related to A177385.

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A193467 E.g.f.: Sum_{n>=0} x^n * exp(n*(n+1)/2*x).

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A177387 E.g.f.: Sum_{n>=0} Product_{k=1..n} sin(k*x).

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A177386 O.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*arcsinh(2x)).

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A193466 E.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} cosh(k*x).

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A249564 a(n) = Sum_{k = 0..n} (k*(k+1)/2)^n.

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A249698 E.g.f.: Sum_{n>=0} Product_{k=1..n} cosh(k*x).

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A249737 E.g.f.: Product_{k=1..n} cosh(k*x).

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A193467 E.g.f.: Sum_{n>=0} x^n * exp(n(n+1)/2x).