A182962 -id:A182962 - cp's OEIS frontend

A088716 G.f. satisfies: A(x) = 1 + xA(x)d/dx[xA(x)] = 1 + xA(x)^2 + x^2A(x)A'(x).

1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970

Offset: 0

Paul D. Hanna, Oct 12 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
M. J. H. Al-Kaabi, Monomial Bases for free pre-Lie algebras, Sem. Lothar. Comb. 71 (2014) B71b.
Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 13.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 11.

Cf. A112916 (A^2), A112911, A112912, A112913, A112914.
Cf. A088715, A182962, A112915, A218222, A238223.
Cf. A300736, A300987, A300989, A300991, A300993.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(j)*a(n-j-1)*(j+1), j=0..n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2017

a=ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = Sum[k*a[[n-k+1]]*a[[k]],{k,1,n}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 21 2014 *)
m = 20; A[_] = 0;
Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
Map[a,Range[20]] (* Oliver Seipel, Nov 03 2024 ,after Schröder 1870 *)

a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*a(k)*a(n-k-1)))

{a(n)=local(G=1+x);for(i=1,n,G=exp(x/(1 - x*deriv(G)/G+x*O(x^n))));polcoeff(log(G)/x,n)} \\ Paul D. Hanna, Jan 01 2011

a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.
Forms column 0 of triangle T=A112911, where the matrix inverse satisfies [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.
Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.
G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.
O.g.f.: A(x) = log(G(x))/x where G(x) is the e.g.f. of A182962 given by G(x) = exp( x/(1 - x*G'(x)/G(x)) ). [Paul D. Hanna, Jan 01 2011]
O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) / A(x) = 0 for n>0. - Paul D. Hanna, May 25 2018
O.g.f. A(x) satisfies [x^n] exp( n * x*A(x) ) * (1 - n*x) = 0 for n>0. - Paul D. Hanna, Jul 24 2019
From Paul D. Hanna, Jul 20 2018 (Start):
O.g.f. A(x) satisfies:
* [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.
* [x^n] exp(-n^(p+1) * x*A(x)) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0. (End)
a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - Vaclav Kotesovec, Feb 21 2014

A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)x^n/n! ) = Sum_{n>=0} a(n)x^n/n! with a(0) = 1.

Original entry on oeis.org

1, 1, 5, 58, 1181, 36696, 1601497, 92969920, 6908883417, 638746871680, 71860612355981, 9664570175364864, 1531263494465900725, 282321785979644121088, 59935663751282958139425, 14517627118656645274771456, 3980008380007702720451029553, 1226189930561023692489563013120

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! + ...
log(A(x)) = x + 2^2*x^2/2! + 3^2*5*x^3/3! + 4^2*58*x^4/4! + 5^2*1181*x^5/5! + ...

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A156325, A156327, A182962, A161968.

nmax = 20; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = Sum[k^2 * Binomial[n-1,k-1]*b[[k]]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b (* Vaclav Kotesovec, Feb 27 2014 *)

{a(n)=if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k^2*a(k-1)*x^k/k!)+x*O(x^n)),n))}

{a(n)=if(n==0,1,sum(k=1,n,k^2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}

seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=1, n, k^2 * binomial(n-1,k-1)*a[k]*a[1+n-k])); a} \\ Andrew Howroyd, Jan 05 2020

a(n) = Sum_{k=1..n} k^2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f.: A(x) = exp( x*A(x) + x^2*A'(x) ). - Paul D. Hanna, Apr 02 2018
E.g.f.: A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x/G(x)) = G(x) is the e.g.f. of A182962, which satisfies:
. G(x) = exp( x/(1 - x*G'(x)/G(x)) );
. a(n) = [x^n/n!] G(x)^(n+1)/(n+1) for n>=0.
a(n) = A161968(n+1)/(n+1), where L(x) = x*exp(x*d/dx L(x)) is the e.g.f. of A161968. - Paul D. Hanna, Feb 21 2014
a(n) ~ c * n * (n!)^2, where c =  A238223 * exp(1) = 0.592451670452494179138706062417512405957... - Vaclav Kotesovec, Feb 27 2014

Terms a(15) and beyond from Andrew Howroyd, Jan 05 2020

A296170 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2) for n>=1.

Original entry on oeis.org

1, 1, -1, -11, -239, -17059, -2145689, -412595231, -111962826751, -40590007936199, -18900753214178609, -10974885891916507219, -7765167486697279401071, -6571694718107813687003051, -6551841491106355785902247049, -7597507878436131044487467850599, -10136619271768255373949409579309439, -15416099624633773180711565727641136271, -26508391106594400233543066679525341764961

Offset: 0

Paul D. Hanna, Dec 07 2017

sign

Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.

			E.g.f.: A(x) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! - 2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! - 40590007936199*x^9/9! - 18900753214178609*x^10/10! - 10974885891916507219*x^11/11! - 7765167486697279401071*x^12/12! - 6571694718107813687003051*x^13/13! - 6551841491106355785902247049*x^14/14! - 7597507878436131044487467850599*x^15/15! +...
To illustrate [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2), form a table of coefficients of x^k in A(x)^(n^2) that begins as
n=1: [(1), (1), -1/2, -11/6, -239/24, -17059/120, -2145689/720, ...];
n=2: [1, (4), (4), -28/3, -196/3, -10472/15, -614264/45, ...];
n=3: [1, 9, (63/2), (63/2), -1701/8, -98217/40, -3168081/80, ...];
n=4: [1, 16, 112, (1232/3), (1232/3), -95648/15, -4835264/45, ...];
n=5: [1, 25, 575/2, 11725/6, (190225/24), (190225/24), ...];
n=6: [1, 36, 612, 6444, 45684, (1043784/5), (1043784/5), ...];
n=7: [1, 49, 2303/2, 102949/6, 4313617/24, 164086349/120, (5086480231/720), (5086480231/720), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^2) by n^2, we obtain the following sequence:
[1, 1, 7, 154, 7609, 695856, 103805719, 23134327168, 7227250033329, 3017857024161280, 1623903877812828871, ..., A296232(n), ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 - 2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 - 15944040740604*x^12 - 1039553309158964*x^13 - 74123498185170292*x^14 - 5736368141560365292*x^15 - 478780244956262592748*x^16 - 42865943103053965559668*x^17 - 4097785410628237071311764*x^18 - 416572537937169684523985420*x^19 - 44873737158384968851319470220*x^20 +...+ A296171(n)*x^n +...

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

Cf. A296171 (log), A296232, A296172, A296174, A296176, A182962.

{a(n) = my(A=[1]); for(i=1,n+1, A=concat(A,0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0,30,print1(a(n),", "))

The logarithm of the e.g.f. A(x) is an integer series:
 log(A(x)) = Sum{n>=1} A296171(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^2) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^2) for n>=1, when evaluated at x = 0.
a(n) ~ c * d^n * n^(2*n-2) / exp(2*n), where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = -0.1875440087... - Vaclav Kotesovec, Dec 23 2017

A300590 E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2) for n>=1.

Original entry on oeis.org

1, 1, 5, 175, 18385, 3759701, 1258735981, 630063839035, 445962163492385, 429694421369414185, 547875295770399220981, 903754519692129905068391, 1892423689107542226463430065, 4948056864672913520114055888445, 15922007799835205487157437619131485, 62245856465769048392433555378169339891, 292266373167286246870149657443033728860481

Offset: 0

Paul D. Hanna, Mar 09 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n^2) = n * [x^(n-1)] exp(x)^(n^2) for n>=1.

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 175*x^3/3! + 18385*x^4/4! + 3759701*x^5/5! + 1258735981*x^6/6! + 630063839035*x^7/7! + 445962163492385*x^8/8! + 429694421369414185*x^9/9! + 547875295770399220981*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [(1), (1), 5/2, 175/6, 18385/24, 3759701/120, 1258735981/720, ...];
n=2: [1, (4), (16), 452/3, 10448/3, 2037388/15, 333368656/45, ...];
n=3: [1, 9, (117/2), (1053/2), 79803/8, 14107743/40, 1472857749/80, ...];
n=4: [1, 16, 160, (4880/3), (78080/3), 11770672/15, 1707161056/45, ...];
n=5: [1, 25, 725/2, 27175/6, (1642225/24), (41055625/24), ...];
n=6: [1, 36, 720, 11340, 180720, (19548324/5), (703739664/5),  ...];
n=7: [1, 49, 2597/2, 154399/6, 11125009/24, (1138996229/120), (205943018701/720), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*(1); 16 = 2^2*(4); 1053/2 = 3^2*(117/2); 78080/3 = 4^2*(4880/3); 41055625/24 = 5^2*(1642225/24); ...
illustrating that: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + 123198985*x^7 + 10931897664*x^8 + 1172808994833*x^9 + 149774206572050*x^10 + ... + A300591(n)*x^n + ...

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

Cf. A182962, A300870, A296170, A300591, A300592, A300594.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^2*V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300591(n)*x^n, a power series in x with integer coefficients.

a(n) ~ c * n!^3 * n^2, where c = 0.1354708370957778563796... - Vaclav Kotesovec, Oct 13 2020

A238223 Decimal expansion of a constant related to A088716.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Feb 21 2014

			0.21795078944715106549282282244231982088...

Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 13.

Cf. A088716, A088715, A158884, A159606, A182962, A193332, A156326, A338377.

Equals lim n->infinity A088716(n)/(n!*n^2).

A300592 E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^3 * [x^(n-1)] A(x)^(n^2) for n>=1.

Original entry on oeis.org

1, 1, 13, 1333, 438073, 328561681, 482408372341, 1262989939509733, 5507311107090685873, 37883505322347710775553, 393149949374099099160049501, 5930998808712507352448964186421, 126060064477829234977371818938653673, 3675839897921109642941288187056728970833, 143727814785299582494066294788162327508528453

Offset: 0

Paul D. Hanna, Mar 09 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n^2) = n * [x^(n-1)] exp(x)^(n^2) for n>=1.

			E.g.f.: A(x) = 1 + x + 13*x^2/2! + 1333*x^3/3! + 438073*x^4/4! + 328561681*x^5/5! + 482408372341*x^6/6! + 1262989939509733*x^7/7! + 5507311107090685873*x^8/8! + 37883505322347710775553*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [(1), (1), 13/2, 1333/6, 438073/24, 328561681/120, ...];
n=2: [1, (4), (32), 2912/3, 228032/3, 167874308/15, ...];
n=3: [1, 9, (189/2), (5103/2), 1468467/8, 1045214163/40, ...];
n=4: [1, 16, 224, (17024/3), (1089536/3), 735471632/15, ...];
n=5: [1, 25, 925/2, 70525/6, (15835225/24), (1979403125/24), ...];
n=6: [1, 36, 864, 23328, 1161792, (654796044/5), (141435945504/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 32 = 2^3*4; 5103/2 = 3^3*189/2; 1089536/3 = 4^3*17024/3; ...
illustrating that: [x^n] A(x)^(n^2) = n^3 * [x^(n-1)] A(x)^(n^2).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 6*x^2 + 216*x^3 + 18016*x^4 + 2718575*x^5 + 667151244*x^6 + 249904389518*x^7 + 136335045655680*x^8 + 104258627494173747*x^9 + 108236370325030253850*x^10 + 148475074256982964816314*x^11 + 263023328027145941803648512*x^12 + ... + A300593(n)*x^n + ...

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

Cf. A182962, A296170, A300590, A300593, A300594.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300593(n)*x^n, a power series in x with integer coefficients.

a(n) ~ c * n!^4, where c = 3.1056678107899395562612789210816... - Vaclav Kotesovec, Oct 14 2020

A300594 E.g.f. A(x) satisfies: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3) for n>=1.

Original entry on oeis.org

1, 1, 9, 1483, 976825, 1507281021, 4409747597401, 21744850191313999, 167557834535988306033, 1913194223179191462419065, 31110747474489521617502800201, 698529144858380953105954686101811, 21123268203104470199318422678044241129, 842759726425517953579189712209822358428213, 43599233739340643789919321494623289001407934105

Offset: 0

Paul D. Hanna, Mar 09 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n^3) = n^2 * [x^(n-1)] exp(x)^(n^3) for n>=1.

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 1483*x^3/3! + 976825*x^4/4! + 1507281021*x^5/5! + 4409747597401*x^6/6! + 21744850191313999*x^7/7! + 167557834535988306033*x^8/8! + 1913194223179191462419065*x^9/9! + 31110747474489521617502800201*x^10/10! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^3) begins:
  n=1: [(1), (1), 9/2, 1483/6, 976825/24, 502427007/40, 4409747597401/720, ...]
  n=2: [1, (8), (64), 6856/3, 1022528/3, 1543097816/15, 2237393526784/45, ...]
  n=3: [1, 27, (945/2), (25515/2), 10692675/8, 14849374869/40, 13978534445001/80, ...]
  n=4: [1, 64, 2304, (226880/3), (14520320/3), 5124803136/5, 20241220116736/45, ...]
  n=5: [1, 125, 16625/2, 2510375/6, (553359625/24), (69169953125/24), ...];
  n=6: [1, 216, 24192, 1918728, 131302080, (56555402904/5), (12215967027264/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 64 = 2^3*8; 25515/2 = 3^3*945/2; 14520320/3 = 4^3*226880/3; ...
illustrating that: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + 4308276119854*x^7 + 4151360558858752*x^8 + 5268077625693186225*x^9 + 8567999843251994553500*x^10 + ... + A300595(n)*x^n + ...

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

Cf. A182962, A296170, A300590, A300592, A300595.

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

E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300595(n)*x^n, a power series in x with integer coefficients.

a(n) ~ c * n!^4 * n^3, where c = 0.40774346023... - Vaclav Kotesovec, Oct 14 2020

A300735 E.g.f. A(x) satisfies: [x^n] A(x)^(2n) = (n+1) [x^(n-1)] A(x)^(2*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 31, 697, 25761, 1371691, 97677343, 8869533681, 993709302337, 134086553693011, 21392941696576671, 3977310371182762153, 851537642070562468321, 207892899850805427254907, 57394298500033495294907551, 17789220343418322663802383841, 6151146653207427022767433596033, 2359535664677835451305256629862051, 999033160522078788619730346474821407

Offset: 0

Paul D. Hanna, Mar 17 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(2*n) = 2 * [x^(n-1)] exp(x)^(2*n) for n>=1.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 697*x^4/4! + 25761*x^5/5! + 1371691*x^6/6! + 97677343*x^7/7! + 8869533681*x^8/8! + 993709302337*x^9/9! + 134086553693011*x^10/10! + ...
such that [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n) for n>=1.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1696*x^4/4! + 60352*x^5/5! + 3134464*x^6/6! + 219316736*x^7/7! + 19655797760*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(2*n) begin:
n=1: [(1), (2), 4, 40/3, 212/3, 7544/15, 195904/45, 13707296/315, ...];
n=2: [1, (4), (12), 128/3, 632/3, 6976/5, 515776/45, 34760896/315, ...];
n=3: [1, 6, (24), (96), 468, 14664/5, 114384/5, 7407552/35, ...];
n=4: [1, 8, 40, (544/3), (2720/3), 82496/15, 1843264/45, 22923136/63, ...];
n=5: [1, 10, 60, 920/3, (4820/3), (9640), 622880/9, 37242080/63, ...];
n=6: [1, 12, 84, 480, 2664, (80448/5), (563136/5), 32495424/35, ...];
n=7: [1, 14, 112, 2128/3, 12572/3, 387128/15, (8018416/45), (64147328/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 2*(1); 12 = 3*(4); 96 = 4*(24); 2720/3 = 5*(544/3); 9640 = 6*(4820/3); 563136/5 = 7*(80448/5); 64147328/45 = 8*(8018416/45); ...
illustrating that: [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - x*A'(x)/A(x)) / (1 - 2*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 4*x^3 + 24*x^4 + 184*x^5 + 1672*x^6 + 17296*x^7 + 198800*x^8 + 2499200*x^9 + 33992000*x^10 + 496281344*x^11 + 7731823616*x^12 + ...

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

Cf. A300736, A182962, A300986, A300988, A300990, A300992.

Cf. A300870, A300590, A296170.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(2*(#A-1))); A[#A] = ((#A)*V[#A-1] - V[#A])/(2*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = my(A=1); for(i=1, n, A = exp( x*(A-x*A')/(A-2*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - x*A'(x)) / (A(x) - 2*x*A'(x)) ).

a(n) ~ c * n!^2 * n^3, where c = 0.008789136598... - Vaclav Kotesovec, Oct 24 2020

A300986 E.g.f. A(x) satisfies: [x^n] A(x)^(3n) = (n + 2) [x^(n-1)] A(x)^(3*n) for n>=1.

Original entry on oeis.org

1, 1, 3, 37, 1009, 44541, 2799931, 233188033, 24562692897, 3168510747769, 488856473079571, 88597562768075901, 18595324838343722833, 4468203984338696710837, 1217521669261709053889739, 373205252376454629490607641, 127806482596653000272128733761, 48605321514711360780713536416753, 20419150659462692416601828820774307, 9431006202634362924849710001022454869

Offset: 0

Paul D. Hanna, Mar 17 2018

nonn

Compare to: [x^n] exp(x)^(3*n) = 3 * [x^(n-1)] exp(x)^(3*n) for n>=1.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 1009*x^4/4! + 44541*x^5/5! + 2799931*x^6/6! + 233188033*x^7/7! + 24562692897*x^8/8! + 3168510747769*x^9/9! + 488856473079571*x^10/10! + ...
such that [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n) for n>=1.
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2/2! + 171*x^3/3! + 4185*x^4/4! + 173583*x^5/5! + 10491039*x^6/6! + 850141575*x^7/7! + 87745941873*x^8/8! + 11141030530395*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in A(x)^(3*n) begins:
n=1: [(1), (3), 15/2, 57/2, 1395/8, 57861/40, 1165671/80, 18892035/112, ...];
n=2: [1, (6), (24), 102, 576, 21834/5, 206244/5, 15974712/35, ...];
n=3: [1, 9, (99/2), (495/2), 11259/8, 401463/40, 7120899/80, 525246849/560, ...];
n=4: [1, 12, 84, (492), (2952), 102708/5, 864756/5, 60722784/35, ...];
n=5: [1, 15, 255/2, 1725/2, (44595/8), (312165/8), 5077035/16, 340795215/112, ...];
n=6: [1, 18, 180, 1386, 9720, (349542/5), (2796336/5), 36178488/7, ...];
n=7: [1, 21, 483/2, 4179/2, 127323/8, 4767147/40, (76271139/80), (686440251/80), ...]; ...
in which the coefficients in parenthesis are related by
3 = 3*(1); 24 = 4*(6); 495/2 = 5*(99/2); 2952 = 6*(492); 312165/8 = 7*(44595/8); 2796336/5 = 8*(349542/5); 686440251/80 = 9*(76271139/80); ...
illustrating that: [x^n] A(x)^(3*n) = (n+2) * [x^(n-1)] A(x)^(3*n).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is an integer power series in x satisfying
log(A(x)) = x * (1 - 2*x*A'(x)/A(x)) / (1 - 3*x*A'(x)/A(x));
explicitly,
log(A(x)) = x + x^2 + 5*x^3 + 36*x^4 + 327*x^5 + 3489*x^6 + 42048*x^7 + 559008*x^8 + 8073243*x^9 + 125328411*x^10 + 2075525505*x^11 + 36460943208*x^12 + ... + A300987(n)*x^n + ...

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

Cf. A300987, A182962, A300735, A300988, A300990, A300992.

Cf. A300870, A300590, A296170.

{a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(3*(#A-1))); A[#A] = ((#A+1)*V[#A-1] - V[#A])/(3*(#A-1)) ); n!*polcoeff( Ser(A), n)}
for(n=0, 25, print1(a(n), ", "))

{a(n) = my(A=1); for(i=1, n, A = exp( x*(A-2*x*A')/(A-3*x*A' +x*O(x^n)) ) ); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies: A(x) = exp( x * (A(x) - 2*x*A'(x)) / (A(x) - 3*x*A'(x)) ).

a(n) ~ c * (n!)^2 * n^5, where c = 0.0001464056080437... - Vaclav Kotesovec, Mar 20 2018

A300870 E.g.f. A(x) satisfies: [x^n] A(x)^(n(n+1)) = n(n+1) * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.

Original entry on oeis.org

1, 1, 7, 307, 37537, 8755561, 3304572391, 1847063377867, 1447456397632897, 1532041772833285777, 2130468278450240803591, 3808068399270998260188451, 8590473242021318921848038817, 24074336129439663228349612217977, 82657249526888437632759608331784807, 343425012928825298349935150449843384891, 1707701025594135213863151839769061397729281

Offset: 0

Paul D. Hanna, Mar 14 2018

nonn

Compare e.g.f. to: [x^n] exp(x)^(n*(n+1)) = (n+1) * [x^(n-1)] exp(x)^(n*(n+1)) for n>=1.

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 307*x^3/3! + 37537*x^4/4! + 8755561*x^5/5! + 3304572391*x^6/6! + 1847063377867*x^7/7! + 1447456397632897*x^8/8! + 1532041772833285777*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n*(n+1)) begins:
n=1: [(1), (2), 8, 328/3, 9728/3, 2241184/15, 420248704/45, ...];
n=2: [1, (6), (36), 432, 11328, 2470464/5, 150254784/5, ...];
n=3: [1, 12, (108), (1296), 29136, 5776128/5, 335166336/5, ...];
n=4: [1, 20, 260, (10480/3), (209600/3), 7265600/3, 1173400640/9, ...];
n=5: [1, 30, 540, 8640, (166800), (5004000), 241367040, 116509893120/7...];
n=6: [1, 42, 1008, 19656, 396816, (53339328/5), (2240251776/5), ...];
n=7: [1, 56, 1736, 124096/3, 2767184/3, 355355392/15, (38932329856/45), (2180210471936/45), ...]; ...
in which the coefficients in parenthesis are related by
2 = 1*2*(1); 36 = 2*3*(6); 1296 = 3*4*(108); 209600/3 = 4*5*(10480/3); 5004000 = 5*6*(166800); 2240251776/5 = 6*7*(53339328/5); ...
illustrating that: [x^n] A(x)^(n*(n+1)) = n*(n+1) * [x^(n-1)] A(x)^(n*(n+1)).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 3*x^2 + 48*x^3 + 1510*x^4 + 71280*x^5 + 4511808*x^6 + 361640832*x^7 + 35516910960*x^8 + 4184770003200*x^9 + ... + A300871(n)*x^n + ...

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

Cf. A300871, A300590, A296170, A182962.

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

a(n) ~ c * n!^3 * n^3, where c = 0.044039511494832369374... - Vaclav Kotesovec, Oct 14 2020

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A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0) = 1.

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A296170 E.g.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2) for n>=1.

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A300590 E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^2 * [x^(n-1)] A(x)^(n^2) for n>=1.

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A238223 Decimal expansion of a constant related to A088716.

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A300592 E.g.f. A(x) satisfies: [x^n] A(x)^(n^2) = n^3 * [x^(n-1)] A(x)^(n^2) for n>=1.

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A300594 E.g.f. A(x) satisfies: [x^n] A(x)^(n^3) = n^3 * [x^(n-1)] A(x)^(n^3) for n>=1.

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A300735 E.g.f. A(x) satisfies: [x^n] A(x)^(2*n) = (n+1) * [x^(n-1)] A(x)^(2*n) for n>=1.

A088716 G.f. satisfies: A(x) = 1 + xA(x)d/dx[xA(x)] = 1 + xA(x)^2 + x^2A(x)A'(x).

A156326 E.g.f.: A(x) = exp( Sum_{n>=1} n^2 * a(n-1)x^n/n! ) = Sum_{n>=0} a(n)x^n/n! with a(0) = 1.

A300735 E.g.f. A(x) satisfies: [x^n] A(x)^(2n) = (n+1) [x^(n-1)] A(x)^(2*n) for n>=1.

A300986 E.g.f. A(x) satisfies: [x^n] A(x)^(3n) = (n + 2) [x^(n-1)] A(x)^(3*n) for n>=1.

A300870 E.g.f. A(x) satisfies: [x^n] A(x)^(n(n+1)) = n(n+1) * [x^(n-1)] A(x)^(n*(n+1)) for n>=1.