A220181 -id:A220181 - cp's OEIS frontend

A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

1, 1, 8, 160, 5888, 345856, 29677568, 3502489600, 544181977088, 107675615297536, 26435436140822528, 7885689342279024640, 2809177794704769548288, 1177952320402008693538816, 574318105367992485583781888, 322156963576521588458420961280, 206009256195720974104252003647488

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
where
A(x) = 1 + tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Cf. A122399, A195415, A220181, A221078, A221079, A224899, A249489, A245322, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

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

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, (exp(2*m*X)-1)^m/(exp(2*m*X)+1)^m); n!*polcoeff(Egf, n)}
for(n=0,20,print1(a(n),", ") )

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

a(n) ~ c * 2^n * (n!)^2 / (sqrt(n) * (log(1+sqrt(2)))^(2*n)), where c = 0.521427744491499132141002572969819345522922990165233786929882335275903215... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

Original entry on oeis.org

1, 1, 8, 164, 6400, 404176, 37541888, 4814990144, 815074508800, 176018678814976, 47223034903789568, 15407438848482919424, 6007522256082907955200, 2758698201106509138251776, 1473586749521302260021198848, 905915791153129699969076117504

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A245322. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 164*x^3/3! + 6400*x^4/4! + 404176*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(2*x)^2 + tan(3*x)^3 + tan(4*x)^4 + tan(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..224

Cf. A122399, A195415, A220181, A221077, A221079, A245322, A224899, A249489, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tan[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[(-1)^((n-k)/2) * 2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

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

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.82830192319144609189890882712268369027077465204866199572119508594067235975..., c = 0.3460492649810724519960613805096579760009441161242336020188358769124140... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A229258 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 3, 31, 573, 18031, 854613, 57433951, 5242645173, 625589806831, 95051257799973, 17976303383444671, 4153215615930529173, 1154304694449774708751, 380809177225169291456133, 147420687475847638142996191, 66303807316628093952943203573

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 31*x^3 + 573*x^4 + 18031*x^5 + 854613*x^6 +...
where
A(x) = 1 + x/(1-x) + 2!*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 573*x^4/4! + 18031*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/4^2 + (exp(9*x)-1)^3/9^3 + (exp(16*x)-1)^4/16^4 + (exp(25*x)-1)^5/25^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..227

Cf. A229257, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Flatten[{1,Table[Sum[(k^2)^(n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=polcoeff(sum(m=0,n,m!*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m^(2*m)),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0, n, (k^2)^(n-k) * k! * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} (k^2)^(n-k) * k! * Stirling2(n, k).

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

A229259 O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 9, 259, 15789, 1693771, 287145789, 71487432619, 24798142070109, 11518873418467051, 6945333793188487869, 5301472723402989073579, 5018547949600497090304029, 5790959348524892656227425131, 8026963462960378548022418765949, 13197920271743736945902641688868139

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 9*x^2 + 259*x^3 + 15789*x^4 + 1693771*x^5 +...
where
A(x) = 1 + x/(1-x) + 2!*2^2*x^2/((1-2^2*1*x)*(1-2^2*2*x)) + 3!*3^3*x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + 4!*4^4*x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 9*x^2/2! + 259*x^3/3! + 15789*x^4/4! + 1693771*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/2^2 + (exp(9*x)-1)^3/3^3 + (exp(16*x)-1)^4/4^4 + (exp(25*x)-1)^5/5^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..201

Cf. A229257, A229258, A229260, A229261, A229233, A229234, A220181, A122399.

Flatten[{1,Table[Sum[k^(2*n-k) * k! * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=polcoeff(sum(m=0,n,m!*m^m*x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/m^m),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0, n, k^(2*n-k) * k! * stirling(n, k, 2))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} k^(2*n-k) * k! * Stirling2(n, k).

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

A245322 E.g.f.: Sum_{n>=0} sin(n*x)^n.

Original entry on oeis.org

1, 1, 8, 161, 6016, 360421, 31628288, 3823725821, 609263681536, 123729353398441, 31195066498285568, 9560281195915697081, 3500145542231863853056, 1508772905238685631514061, 756360258034794813559144448, 436312320288025061112662937941, 286966475921556619941746443288576

Offset: 0

Vaclav Kotesovec, Nov 05 2014

nonn

It appears that for n >= 1, a(2*n) is even and a(2*n-1) is odd. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A224899 and A221078. - Peter Bala, May 31 2022

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

Cf. A122399, A195415, A220181, A221077, A221078, A224899, A249489, A338040.

nmax=20; Flatten[{1,Rest[CoefficientList[Series[Sum[Sin[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]}]
Flatten[{1,Table[Sum[(-1)^k * (n-2*k)^n * 2^(2*k-n) * Sum[Binomial[n-2*k,j] * (-1)^j * (n-2*k-2*j)^n,{j,0,n-2*k}],{k,0,n/2}],{n,1,20}]}]

{a(n)=n!*polcoeff(sum(k=0, n, sin(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.6508143537621057095493599669955786931108630276472035393383790812849064745..., c = 0.447880926276318254580767843378566025547642779941081708311676940459098... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A187755 E.g.f.: Sum_{n>=0} (1 - exp(-n^2*x))^n.

Original entry on oeis.org

1, 1, 31, 3991, 1340251, 929043391, 1153715889691, 2333670966674671, 7180487882511523051, 31919495229412870788031, 196909477461357591810377851, 1632140626754602443266222263951, 17701927686793740884448652685728651, 245721757633690118910277310669218472671

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			O.g.f.: F(x) = 1 + x + 31*x^2 + 3991*x^3 + 1340251*x^4 + 929043391*x^5 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^6*3!*x^3/((1+3^2*1*x)*(1+3^2*2*x)*(1+3^2*3*x)) + 4^8*4!*x^4/((1+4^2*1*x)*(1+4^2*2*x)*(1+4^2*3*x)*(1+4^2*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 31*x^2/2! + 3991*x^3/3! + 1340251*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^2*x))^3 + (1-exp(-4^2*x))^4 + (1-exp(-5^2*x))^5 + (1-exp(-6^2*x))^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..162

Cf. A220181, A203798, A229260.

Flatten[{1,Table[Sum[(-1)^(n-k) * k^(2*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^2*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, m^(2*m)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

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

O.g.f.: Sum_{n>=0} n^(2*n) * n! * x^n / Product_{k=1..n} (1 + n^2*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n) * k! * Stirling2(n,k).
a(n) == 1 (mod 10) for n>=0.
a(n) == 31 (mod 60) for n>=2.
a(n) ~ c * d^n * (n!)^3 / n, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.192038502554748256318271067254582378566365276592... . - Vaclav Kotesovec, May 08 2014

A195415 E.g.f.: Sum_{n>=1} tanh(nx)^n = Sum_{n>=1} a(n)4^(n-1)*x^n/n!.

Original entry on oeis.org

1, 2, 10, 92, 1351, 28982, 855100, 33214232, 1642999501, 100843185962, 7520379392890, 669760178257172, 70211429619908851, 8558006664633638942, 1200128210993564085880, 191861070874818576596912, 34685967730611200643509401, 7041037426518318365605795922

Offset: 1

Paul D. Hanna, Sep 17 2011

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, ...], with an apparent period of 6. - Peter Bala, May 29 2022

			E.g.f.: A(x) = x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
or, equivalently,
A(x) = x + 2*4*x^2/2! + 10*4^2*x^3/3! + 92*4^3*x^4/4! + 1351*4^4*x^5/5! +...
where
A(x) = tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 1..262

Cf. A122399, A220181, A221077, A221078, A224899, A245322, A338040.

seq(coeff(n!/4^(n-1)*series(add(tanh(n*x)^n, n = 1..100), x, 101), x, n), n = 1..100); # Peter Bala, May 29 2022

nmax = 20; Rest[CoefficientList[Series[Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! / 4^(Range[0, nmax] - 1)] (* Vaclav Kotesovec, May 31 2022 *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=1, n, tanh(m*X)^m); n!/4^(n-1)*polcoeff(Egf, n)}

E.g.f.: Sum_{n>=1} ( 1 - 2/(1+exp(2*n*x)) )^n = Sum_{n>=1} a(n)*4^(n-1)/n!.
a(n) ~ c * d^n * n^(2*n + 1/2), where d = 1 / (2 * exp(2) * log(1+sqrt(2))^2) = 0.0871085887239583895519632137900851584739951067757899616766024190... and c = 13.10490857177911562030370300610447966745088413236135355214718... - Vaclav Kotesovec, May 31 2022
a(n) = A221077(n) / 4^(n-1). - Vaclav Kotesovec, Jun 02 2022

A203798 E.g.f.: Sum_{n>=0} (1 - exp(-n^3*x))^n.

Original entry on oeis.org

1, 1, 127, 115027, 383578651, 3406562690251, 66363706916031547, 2491358400855491082427, 164269869314849711368915051, 17742882813152530090093631133451, 2973340319455184373850280909330520667, 740141055495168376026146815923984436993627

Offset: 0

Paul D. Hanna, Jan 09 2013

nonn

			O.g.f.: F(x) = 1 + x + 127*x^2 + 115027*x^3 + 383578651*x^4 +...
where
F(x) = 1 + x/(1+x) + 2^6*2!*x^2/((1+2^3*1*x)*(1+2^3*2*x)) + 3^9*3!*x^3/((1+3^3*1*x)*(1+3^3*2*x)*(1+3^3*3*x)) + 4^12*4!*x^4/((1+4^3*1*x)*(1+4^3*2*x)*(1+4^3*3*x)*(1+4^3*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 127*x^2/2! + 115027*x^3/3! + 383578651*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^3*x))^2 + (1-exp(-3^3*x))^3 + (1-exp(-4^3*x))^4 + (1-exp(-5^3*x))^5 + (1-exp(-6^3*x))^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A220181, A187755, A242229.

Flatten[{1,Table[Sum[(-1)^(n-k) * k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^3*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, m^(3*m)*m!*x^m/prod(k=1, m, 1+m^3*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, (-1)^(n-k)*k^(3*n)*k!*stirling(n, k, 2))}
for(n=0, 20, print1(a(n), ", "))

O.g.f.: Sum_{n>=0} n^(3*n) * n! * x^n / Product_{k=1..n} (1 + n^3*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(3*n) * k! * Stirling2(n,k).
a(n) == 1 (mod 6) for n>=0.
a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 19.56009813649729638637945621039407270230134093295681610091365833339628... and c = 0.068225708245494700607840778486121251627676796315... . - Vaclav Kotesovec, May 08 2014

A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

Original entry on oeis.org

1, 15, 1267, 316275, 174397531, 179770837155, 310789895286907, 834906367019076675, 3293344593080631993211, 18259284528276047000517795, 137429981152689382429349060347, 1365009985652048448232840864764675, 17475885712645599218827214639383437691

Offset: 1

Paul D. Hanna, Dec 06 2012

nonn

Compare to the trivial identity: x = Sum_{n>=1} (1 - exp(-x))^n/n.

Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.

			E.g.f.: A(x) = x + 15*x^2/2! + 1267*x^3/3! + 316275*x^4/4! + 174397531*x^5/5! +...
where
A(x) = (1-exp(-x)) + (1-exp(-4*x))^2/2 + (1-exp(-9*x))^3/3 + (1-exp(-16*x))^4/4 + (1-exp(-25*x))^5/5 +...

Seiichi Manyama, Table of n, a(n) for n = 1..162

Cf. A092552, A187755, A220181, A242228.

Table[Sum[(-1)^(n-k) * k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, May 08 2014 *)

a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k^2*x+x*O(x^n)))^k/k), n)
for(n=1,20,print1(a(n),", "))

a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

{a(n)=sum(k=1, n, (-1)^(n-k)*k^(2*n-1)*k!*stirling(n, k, 2))}
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

O.g.f.: Sum_{n>=1} n^(2*n-1) * n! * x^n / Product_{k=1..n} (1 - n^2*k*x). - Paul D. Hanna, Jan 05 2013
a(n) = Sum_{k=1..n} (-1)^(n-k) * k^(2*n-1) * k! * Stirling2(n,k). - Paul D. Hanna, Jan 05 2013
a(n) ~ c * d^n * (n!)^3 / n^2, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.175744118254830086361220160145768507562830495967... . - Vaclav Kotesovec, May 08 2014

A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2kx).

Original entry on oeis.org

1, 1, 2, 14, 168, 3147, 90563, 3561231, 185790622, 12599020184, 1071164190670, 111813313594259, 14140296360430353, 2132273568722682621, 378197030144360862958, 78127192632748956075174, 18627308660113953164384812, 5081218748742336002185874439, 1574128413278644602881499193579

Offset: 0

Paul D. Hanna, Sep 17 2013

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 168*x^4 + 3147*x^5 + 90563*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/((1-2^2*1*x)*(1-2^2*2*x)) + x^3/((1-3^2*1*x)*(1-3^2*2*x)*(1-3^2*3*x)) + x^4/((1-4^2*1*x)*(1-4^2*2*x)*(1-4^2*3*x)*(1-4^2*4*x)) +...
Exponential Generating Function.
E.g.f.: E(x) = 1 + x + 2*x^2/2! + 14*x^3/3! + 168*x^4/4! + 3147*x^5/5! +...
where
E(x) = 1 + (exp(x)-1) + (exp(4*x)-1)^2/(2!*4^2) + (exp(9*x)-1)^3/(3!*9^3) + (exp(16*x)-1)^4/(4!*16^4) + (exp(25*x)-1)^5/(5!*25^5) +...

Seiichi Manyama, Table of n, a(n) for n = 0..242

Cf. A229258, A229259, A229260, A229261; A229233, A229234, A220181, A122399.

Flatten[{1,Table[Sum[(k^2)^(n-k) * StirlingS2[n, k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-m^2*k*x +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=n!*polcoeff(sum(m=0,n,(exp(m^2*x+x*O(x^n))-1)^m/(m!*m^(2*m))),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=sum(k=0, n, (k^2)^(n-k) * stirling(n, k, 2))}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} (k^2)^(n-k) * Stirling2(n, k).

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

cp's OEIS Frontend

A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

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A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

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A229258 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2*k*x).

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A229259 O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2*k*x).

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A195415 E.g.f.: Sum_{n>=1} tanh(n*x)^n = Sum_{n>=1} a(n)*4^(n-1)*x^n/n!.

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A229258 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 - n^2kx).

A229259 O.g.f.: Sum_{n>=0} n! * n^n * x^n / Product_{k=1..n} (1 - n^2kx).

A195415 E.g.f.: Sum_{n>=1} tanh(nx)^n = Sum_{n>=1} a(n)4^(n-1)*x^n/n!.

A229257 O.g.f.: Sum_{n>=0} x^n / Product_{k=1..n} (1 - n^2kx).