A226750 -id:A226750 - cp's OEIS frontend

A295813 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172.

1, 3, 48, 3271, 575163, 185377116, 93039467356, 66505075585875, 63970743282062646, 79580632411431634441, 124299284968805234137968, 238188439678208173206500760, 549611050835556942751087049225, 1503700734638162443238902233252144, 4814751647416985610768723994195186728, 17841762828286483988438913318683740082187, 75777421917902616009655480827109144353730842

Offset: 1

Paul D. Hanna, Dec 09 2017

nonn

E.g.f. G(x) of A296172 satisfies: [x^(n-1)] G(x)^(n^3) = [x^n] G(x)^(n^3) for n>=1.

			G.f.: A(x) = x + 3*x^2 + 48*x^3 + 3271*x^4 + 575163*x^5 + 185377116*x^6 + 93039467356*x^7 + 66505075585875*x^8 + 63970743282062646*x^9 + 79580632411431634441*x^10 + 124299284968805234137968*x^11 + 238188439678208173206500760*x^12 +...
The series reversion equals the logarithm of the e.g.f. of A296172, which begins:
Series_Reversion(A(x)) = x - 3*x^2 - 30*x^3 - 2686*x^4 - 517311*x^5 - 173118807*x^6 - 88535206152*x^7 - 63977172334344*x^8 - 61971659588102940*x^9 - 77470793599569049440*x^10 - 121439997599825393413344*x^11 - 233353875172602479932391040*x^12 +...+ A296173(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A296172, A296173, A295812, A295814.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^3)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^3 ); polcoeff(serreverse(log(Ser(A))), n)}
for(n=1, 30, print1(a(n), ", "))

G.f. is the series reversion of the logarithm of the e.g.f. of A296172.

a(n) ~ sqrt(1-c) * 3^(3*n - 3) * n^(2*n - 7/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n - 3) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = -A226750. - Vaclav Kotesovec, Oct 13 2020

A317337 O.g.f. A(x) satisfies: [x^n] exp( n^3xA(x) ) * (n+1 - n*A(x)) = 0 for n>=1.

Original entry on oeis.org

1, 1, 12, 729, 111440, 31377625, 14001201036, 9064452341847, 8027821828474816, 9322437359885669613, 13746212321035446900300, 25094943743950232692612534, 55574014665416527079564569056, 146797467684802516650481763597455, 456012687037844090869850529901126900, 1645914373011657806464530612985244787000

Offset: 0

Paul D. Hanna, Aug 01 2018

nonn

Compare: the factorial series F(x) = Sum_{n>=0} n!*x^n satisfies
(1) [x^n] exp( n^2*x*F(x) ) * (n + 1 - n*F(x)) = 0 for n>=1,
(2) [x^n] exp( n^3*x*F(x) ) * (n^2 + 1 - n^2*F(x)) = 0 for n>=1.
It is remarkable that this sequence should consist entirely of integers.

			O.g.f.: A(x) = 1 + x + 12*x^2 + 729*x^3 + 111440*x^4 + 31377625*x^5 + 14001201036*x^6 + 9064452341847*x^7 + 8027821828474816*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^3*x*A(x) ) * (n+1 - n*A(x)) begins:
n=1: [1, 0, -23, -4376, -2674995, -3765464504, ...];
n=2: [1, 6, 0, -8908, -5494464, -7640806512, ...];
n=3: [1, 24, 549, 0, -8632395, -12056269968, ...];
n=4: [1, 60, 3616, 204712, 0, -17114998496, -45010750350080, ...];
n=5: [1, 120, 14505, 1750880, 197597325, 0, -60559334101475, ...];
n=6: [1, 210, 44352, 9406044, 1987128000, 391935493296, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A305110, A305114, A305115, A305116, A317338.

{a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^3*x*(Ser(A)) ) * (m - (m-1)*Ser(A)) )[m]/(m-1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ sqrt(1-c) * 3^(3*n) * n^(2*n - 3/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 06 2018

A317345 E.g.f. A(x) satisfies: [x^n] exp(n^3*x) / A(x)^(n^2) = 0 for n >= 1.

Original entry on oeis.org

1, 1, 5, 445, 196105, 221673401, 501981700621, 1983064113021685, 12488526496641458705, 117611695946767352571505, 1578802193598207376026165781, 29098684071572000208903027320621, 714476480265312671332820625804579865, 22796869288656035590303941174243615386665, 925701505348044648968634173494720540556875805

Offset: 0

Paul D. Hanna, Jul 26 2018

nonn

It is remarkable that the logarithm of the e.g.f. A(x) should be an integer series.

Periodic modulo 10: a(5*n+k) = [1,1,5,5,5](k) (mod 10), for n>=0 and k = 0..4 (conjecture).

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + 12488526496641458705*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x)^(n^2) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ... + A317346(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A317343, A317346.

{a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); n!*A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ sqrt(1-c) * 3^(3*n - 7/3) * n^(3*n - 2) / (exp(3*n) * c^(n - 1/3) * (3-c)^(2*n - 2)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 07 2018

A274712 a(n) = A008277(3n-1,n) / (n(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.

Original entry on oeis.org

1, 5, 161, 14575, 2671669, 833607138, 397984073059, 270609861663900, 248922595132336125, 298037055910658382175, 450755158919281716609746, 840770855566250627155136090, 1896671776639253430025972662743, 5091278095597325836977485757711800, 16040729445423172146341201903726496024, 58625927208516621021861960954787323034320, 246047331971247756894582227572712664877434765, 1175344062721738572130662103242054758238706829325

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A274713.

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

{a(n) = abs( stirling(3*n-1, n, 2) / (n*(n+1)/2) )}
for(n=1, 20, print1(a(n), ", "))

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

a(n) = A274713(n) / (n*(n+1)/2), where A274713(n) is the number of partitions of a {3*n-1}-set into n nonempty subsets.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1) / (n*(n+1)/2).
a(n) ~ sqrt(2) * 3^(3*n-1) * n^(2*n-7/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211... = -A226750. - Vaclav Kotesovec, Jul 06 2016

A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 15, 966, 145750, 40075035, 17505749898, 11143554045652, 9741955019900400, 11201516780955125625, 16392038075086211019625, 29749840488672593296243236, 65580126734167548918100615020, 172597131674172062132363512309613, 534584200037719212882636004559739000, 1924887533450780657560944228447179522880, 7973126100358260458973226689851075932667520, 37645241791600906804871080818625037726247519045

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

a(n) is divisible by the triangular numbers: a(n) / (n*(n+1)/2) = A274712(n).

			O.g.f.: A(x) = x + 15*x^2 + 966*x^3 + 145750*x^4 + 40075035*x^5 + 17505749898*x^6 + 11143554045652*x^7 + 9741955019900400*x^8 +...
where
A(x) = exp(-x)*x + 2^5*exp(-2^3*x)*x^2/2! + 3^8*exp(-3^3*x)*x^3/3! + 4^11*exp(-4^3*x)*x^4/4! + 5^14*exp(-5^3*x)*x^5/5! + 6^17*exp(-6^3*x)*x^6/6! + 7^20*exp(-7^3*x)*x^7/7! + 8^23*exp(-8^3*x)*x^8/8! +...+ n^(3*n-1)*exp(-n^3*x)*x^n/n! +...
simplifies to an integer series.

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A129506, A217913, A274712, A008277.

Table[StirlingS2[3*n - 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Jul 06 2016 *)

{a(n) = abs( stirling(3*n-1, n, 2) )}
for(n=1, 20, print1(a(n), ", "))

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

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

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

O.g.f.: Sum_{n>=1} n^(3*n-1) * exp(-n^3*x) * x^n / n!, an integer series.
a(n) = A008277(3*n-1,n) for n>=1, where A008277 are the Stirling numbers of the second kind.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1).
a(n) = [x^(2*n-1)] 1 / Product_{k=1..n} (1 - k*x).
a(n) ~ 3^(3*n-1) * n^(2*n-3/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(2*Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968... = -A226750. - Vaclav Kotesovec, Jul 06 2016

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A295813 G.f. A(x) satisfies: G(A(x)) = exp(x), where G(x) equals the e.g.f. of A296172.

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A317337 O.g.f. A(x) satisfies: [x^n] exp( n^3*x*A(x) ) * (n+1 - n*A(x)) = 0 for n>=1.

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A317345 E.g.f. A(x) satisfies: [x^n] exp(n^3*x) / A(x)^(n^2) = 0 for n >= 1.

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A274712 a(n) = A008277(3*n-1,n) / (n*(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.

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A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

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A317337 O.g.f. A(x) satisfies: [x^n] exp( n^3xA(x) ) * (n+1 - n*A(x)) = 0 for n>=1.

A274712 a(n) = A008277(3n-1,n) / (n(n+1)/2) for n>=1, where A008277 are the Stirling numbers of the second kind.