A088715 -id:A088715 - 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

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).

A352236 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 2x*A'(x)).

Original entry on oeis.org

1, 1, 3, 19, 185, 2353, 36075, 638115, 12683761, 278485217, 6674259667, 173097575603, 4826128088489, 143896870347793, 4568544366818747, 153883892657000259, 5481761893234193889, 205939077652874352577, 8138639816942009694627, 337568614331296733526867

Offset: 0

Paul D. Hanna, Mar 08 2022

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 185*x^4 + 2353*x^5 + 36075*x^6 + 638115*x^7 + 12683761*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(2*n+1) begins:
n=0: [1,  1,   3,   19,   185,   2353,   36075, ...];
n=1: [1,  3,  12,   76,   705,   8595,  127680, ...];
n=2: [1,  5,  25,  165,  1490,  17506,  252050, ...];
n=3: [1,  7,  42,  294,  2632,  30016,  419454, ...];
n=4: [1,  9,  63,  471,  4239,  47295,  643017, ...];
n=5: [1, 11,  88,  704,  6435,  70785,  939312, ...];
n=6: [1, 13, 117, 1001,  9360, 102232, 1329016, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1), n >= 1,
as illustrated by
[x^1] A(x)^3 = 3 = [x^0] 3*A(x)^3 = 3*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^7 = 294 = [x^2] 7*A(x)^7 = 7*42;
[x^4] A(x)^9 = 4239 = [x^3] 9*A(x)^9 = 9*471;
[x^5] A(x)^11 = 70785 = [x^4] 11*A(x)^11 = 11*6435;
[x^6] A(x)^13 = 1329016 = [x^5] 13*A(x)^13 = 13*102232; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^2) = 1 + x + 5*x^2 + 42*x^3 + 471*x^4 + 6435*x^5 + 102232*x^6 + 1837630*x^7 + ... + A317352(n)*x^n + ...
where B(x)^2 = (1/x) * Series_Reversion( x/A(x)^2 ).

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A088715, A286797, A317352, A352235, A352237, A352238.

/* Using A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 2*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

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

G.f. A(x) satisfies:
(1) [x^n] A(x)^(2*n+1) = [x^(n-1)] (2*n+1) * A(x)^(2*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 2*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (2*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(2*x) dx ).
a(n) ~ c * 2^n * n! * n^(3/2), where c = 0.06926688933886004638602492... - Vaclav Kotesovec, Nov 16 2023

A352235 G.f. A(x) satisfies: A(x) = 1 + xA(x) / (A(x) - 3x*A'(x)).

Original entry on oeis.org

1, 1, 3, 24, 309, 5262, 108894, 2618718, 71246145, 2154788970, 71563126710, 2586270267600, 100995812044266, 4237522832234832, 190126298040192912, 9085093650185205498, 460711407231295513689, 24715373661154672634058, 1398648334415007990887454

Offset: 0

Paul D. Hanna, Mar 08 2022

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 309*x^4 + 5262*x^5 + 108894*x^6 + 2618718*x^7 + 71246145*x^8 + ...
such that A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+2) begins:
n=0: [1,  2,   7,   54,   675,  11286,   230742, ...];
n=1: [1,  5,  25,  190,  2210,  34981,   688635, ...];
n=2: [1,  8,  52,  416,  4642,  69872,  1322848, ...];
n=3: [1, 11,  88,  759,  8349, 120549,  2195886, ...];
n=4: [1, 14, 133, 1246, 13790, 193060,  3391017, ...];
n=5: [1, 17, 187, 1904, 21505, 295154,  5017618, ...];
n=6: [1, 20, 250, 2760, 32115, 436524,  7217250, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2), n >= 1,
as illustrated by
[x^1] A(x)^2 = 2 = [x^0] 2*A(x)^2 = 2*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^8 = 416 = [x^2] 8*A(x)^8 = 8*52;
[x^4] A(x)^11 = 8349 = [x^3] 11*A(x)^11 = 11*759;
[x^5] A(x)^14 = 193060 = [x^4] 14*A(x)^14 = 14*13790;
[x^6] A(x)^17 = 5017618 = [x^5] 17*A(x)^17 = 17*295154; ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..378

Cf. A088715, A286797, A317352, A352236, A352237, A352238.

/* Using A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))

/* Using [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2)*A(x)^(3*n+2) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A-2)+2) - Ser(A)^(3*(#A-2)+2)/(3*(#A-2)+2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

G.f. A(x) satisfies:
(1) [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2) for n >= 1.
(2) A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
(3) A'(x) = A(x) * (1 + x/(1 - A(x))) / (3*x).
(4) A(x) = exp( Integral (1 + x/(1 - A(x))) / (3*x) dx ).
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.09232038797888963484135336... - Vaclav Kotesovec, Nov 16 2023

A352237 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 3x*A'(x)).

Original entry on oeis.org

1, 1, 4, 37, 532, 9994, 226252, 5910445, 173581060, 5634589906, 199792389160, 7671942375898, 316936631324368, 14011781050744984, 660054967923455212, 33008607551445324157, 1746771084107236755604, 97536010045722766992778, 5731874036042145864368824

Offset: 0

Paul D. Hanna, Mar 08 2022

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 37*x^3 + 532*x^4 + 9994*x^5 + 226252*x^6 + 5910445*x^7 + 173581060*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+1) begins:
n=0: [1,  1,   4,   37,   532,   9994,   226252, ...];
n=1: [1,  4,  22,  200,  2717,  48788,  1069122, ...];
n=2: [1,  7,  49,  462,  6069, 104664,  2219784, ...];
n=3: [1, 10,  85,  850, 11020, 183832,  3777355, ...];
n=4: [1, 13, 130, 1391, 18083, 294203,  5869734, ...];
n=5: [1, 16, 184, 2112, 27852, 445632,  8659920, ...];
n=6: [1, 19, 247, 3040, 41002, 650161, 12353059, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1), n >= 1,
as illustrated by
[x^1] A(x)^4 = 4 = [x^0] 4*A(x)^4 = 4*1;
[x^2] A(x)^7 = 49 = [x^1] 7*A(x)^7 = 7*7;
[x^3] A(x)^10 = 850 = [x^2] 10*A(x)^10 = 10*85;
[x^4] A(x)^13 = 18083 = [x^3] 13*A(x)^13 = 13*1391;
[x^5] A(x)^16 = 445632 = [x^4] 16*A(x)^16 = 16*27852;
[x^6] A(x)^19 = 12353059 = [x^5] 19*A(x)^19 = 19*650161; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^3) = 1 + x + 7*x^2 + 85*x^3 + 1391*x^4 + 27852*x^5 + 650161*x^6 + 17204220*x^7 + ...
where B(x)^3 = (1/x) * Series_Reversion( x/A(x)^3 ).

Vaclav Kotesovec, Table of n, a(n) for n = 0..378

Cf. A088715, A286797, A317352, A352235, A352236, A352238.

/* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))

/* Using [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1)*A(x)^(3*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A)-2) - Ser(A)^(3*(#A)-2)/(3*(#A)-2)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

G.f. A(x) satisfies:
(1) [x^n] A(x)^(3*n+1) = [x^(n-1)] (3*n+1) * A(x)^(3*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (3*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(3*x) dx ).
a(n) ~ c * 3^n * n! * n^(4/3), where c = 0.0543186200722307001992331... - Vaclav Kotesovec, Nov 16 2023

A352238 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 4x*A'(x)).

Original entry on oeis.org

1, 1, 5, 61, 1161, 28857, 864141, 29861749, 1160382737, 49854838897, 2340623599381, 119051103325613, 6516915195123097, 381912592990453545, 23856225840952434333, 1582482450123627473637, 111113139625779846025761, 8234335766045466358238433

Offset: 0

Paul D. Hanna, Mar 08 2022

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 61*x^3 + 1161*x^4 + 28857*x^5 + 864141*x^6 + 29861749*x^7 + 1160382737*x^8 + ...
such that A(x) = 1 + x*A(x)^2/(A(x) - 4*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(4*n+1) begins:
n=0: [1,  1,   5,   61,   1161,   28857,   864141, ...];
n=1: [1,  5,  35,  415,   7430,  176286,  5107530, ...];
n=2: [1,  9,  81,  993,  17127,  389583, 10916559, ...];
n=3: [1, 13, 143, 1859,  31564,  693212, 18802212, ...];
n=4: [1, 17, 221, 3077,  52309, 1118549, 29427153, ...];
n=5: [1, 21, 315, 4711,  81186, 1704906, 43640030, ...];
n=6: [1, 25, 425, 6825, 120275, 2500555, 62513875, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1) * A(x)^(4*n+1), n >= 1,
as illustrated by
[x^1] A(x)^5 = 5 = [x^0] 5*A(x)^5 = 5*1;
[x^2] A(x)^9 = 81 = [x^1] 9*A(x)^9 = 9*9;
[x^3] A(x)^13 = 1859 = [x^2] 13*A(x)^13 = 13*143;
[x^4] A(x)^17 = 52309 = [x^3] 17*A(x)^17 = 17*3077;
[x^5] A(x)^21 = 1704906 = [x^4] 21*A(x)^21 = 21*81186;
[x^6] A(x)^25 = 62513875 = [x^5] 25*A(x)^25 = 25*2500555; ...
Also, compare the above terms along the diagonal to the series
B(x) = A(x*B(x)^4) = 1 + x + 9*x^2 + 143*x^3 + 3077*x^4 + 81186*x^5 + 2500555*x^6 + 87388600*x^7 + ...
where B(x)^4 = (1/x) * Series_Reversion( x/A(x)^4 ).

Vaclav Kotesovec, Table of n, a(n) for n = 0..363

Cf. A088715, A286797, A317352, A352235, A352236, A352237.

/* Using A(x) = 1 + x*A(x)^2/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1,n, A = 1 + x*A^2/(A - 4*x*A' + x*O(x^n)) );
polcoeff(A,n)}
for(n=0,20, print1(a(n),", "))

/* Using [x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1)*A(x)^(4*n+1) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff((x*Ser(A)^(4*(#A)-3) - Ser(A)^(4*(#A)-3)/(4*(#A)-3)),#A-1));A[n+1]}
for(n=0,20, print1(a(n),", "))

G.f. A(x) satisfies:
(1) [x^n] A(x)^(4*n+1) = [x^(n-1)] (4*n+1) * A(x)^(4*n+1) for n >= 1.
(2) A(x) = 1 + x*A(x)^2/(A(x) - 4*x*A'(x)).
(3) A'(x) = A(x) * (1 + x*A(x)/(1 - A(x))) / (4*x).
(4) A(x) = exp( Integral (1 + x*A(x)/(1 - A(x)))/(4*x) dx ).
a(n) ~ c * 4^n * n! * n^(5/4), where c = 0.0440035900116077498469559... - Vaclav Kotesovec, Nov 16 2023

A158884 G.f. A(x) satisfies: d/dx xA(x) = 1+x + x[d/dx log(A(x))].

Original entry on oeis.org

1, 1, -1, 4, -23, 166, -1410, 13602, -145803, 1711690, -21785618, 298370920, -4372151566, 68234087624, -1129894265272, 19788479904366, -365520041466291, 7103187300763530, -144897616964143050, 3096285550330959336

Offset: 0

Paul D. Hanna, Apr 30 2009

sign

			G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
where coefficients in parenthesis form A158883 and signed A088716
and A(x)^-1 (first row) is the g.f. of signed A088715.

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

Cf. A158883, A088715, A088716.

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);Vec(Ser(A)^(n+1)/(n+1))[n+1]}

G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017

A295766 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2 with A'(0) = 1.

Original entry on oeis.org

1, 1, 5, 90, 3204, 170987, 12162683, 1087504130, 118227836360, 15304211345298, 2324856843115770, 409872125913866852, 83092182794794380856, 19214014336799266619671, 5030971580159960051721815, 1481724835890098667273954338, 487883202104697456579537247232, 178595806151469762148235569612814, 72312528698655521190143801630975174

Offset: 0

Paul D. Hanna, Jan 31 2018

nonn

Compare g.f. to: [x^(n-1)] G(x)^(n^2)/n^2 = [x^(n-2)] G(x)^(n^2)/(n-1) for n>=2 holds when G(x) = exp(x).

			G.f.: A(x) = 1 + x + 5*x^2 + 90*x^3 + 3204*x^4 + 170987*x^5 + 12162683*x^6 + 1087504130*x^7 + 118227836360*x^8 + 15304211345298*x^9 + 2324856843115770*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 5, 90, 3204, 170987, 12162683, ...];
n=2: [1, 4, 26, 424, 14107, 729196, 50993674, ...];
n=3: [1, 9, 81, 1254, 37602, 1833597, 124332453, ...];
n=4: [1, 16, 200, 3200, 86084, 3846720, 248466736, ...];
n=5: [1, 25, 425, 7550, 188750, 7566705, 455263225, ...];
n=6: [1, 36, 810, 16680, 410499, 14777964, 808802730, ...];
n=7: [1, 49, 1421, 34594, 886312, 29473255, 1444189495, ...]; ...
in which the main diagonal
[1, 4, 81, 3200, 188750, 14777964, 1444189495, ...]
is related to an adjacent diagonal by dividing by n^2 like so:
[1, 4/4, 81/9, 3200/16, 188750/25, 14777964/36, 1444189495/49, ...]
= [1, 1, 9, 200, 7550, 410499, 29473255, ...].
Thus [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2.

Paul D. Hanna, Table of n, a(n) for n = 0..260

Cf. A088715, A295811, A075427.

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

/* Informal method of obtaining N terms: */
N=30; A=[1]; for(n=2,N, A=concat(A,0); V=Vec(Ser(A)^(n^2)); A[#A] = V[#A-1] - V[#A]/n^2 );A

a(A075427(k) - 1) is odd for n>=0 and a(n) is even elsewhere (conjecture).

A295811 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = 2n [x^(n-2)] A(x)^(n^2) for n>=2, with A(0) = 1.

Original entry on oeis.org

1, 1, 2, 11, 140, 2898, 80844, 2786091, 113184008, 5266198778, 275248731860, 15939117549502, 1012084698990904, 69901132180300132, 5217426460077854712, 418615099531669351443, 35942031310982080239120, 3289533291926922095871546, 319841125714352173292953668, 32937612567848507536114539402, 3582858531960091228861488651864

Offset: 0

Paul D. Hanna, Feb 02 2018

nonn

Compare g.f. to: [x^(n-1)] G(x)^n = 2 * [x^(n-2)] G(x)^n for n>=2 holds when G(x) = 1/(1-x).

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 2898*x^5 + 80844*x^6 + 2786091*x^7 + 113184008*x^8 + 5266198778*x^9 + 275248731860*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 2, 11, 140, 2898, 80844, ...];
n=2: [1, 4, 14, 72, 741, 13724, 364546, ...];
n=3: [1, 9, 54, 327, 2826, 42660, 1017720, ...];
n=4: [1, 16, 152, 1216, 10540, 129376, 2559792, ...];
n=5: [1, 25, 350, 3775, 37750, 427480, 6820800, ...];
n=6: [1, 36, 702, 10056, 123165, 1477980, 20712546, ...];
n=7: [1, 49, 1274, 23667, 359856, 4953998, 69355972, ...]; ...
in which the main diagonal
D0 = [1, 4, 54, 1216, 37750, 1477980, 69355972, 3775816704, ...]
and the adjacent diagonal
D1 = [1, 9, 152, 3775, 123165, 4953998, 235988544, 12954335103, ...]
are related by D0[n-1] = 2*n*D1[n-2] for n>=2.
The related sequence D0[n-1]/n^2, n>=1, begins:
[1, 1, 6, 76, 1510, 41055, 1415428, 58997136, 2878741134, 160698224230, ...].

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

Cf. A295766, A088715, A300873.

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

a(2^k - 1) is odd for k>=0 and a(n) is even elsewhere (conjecture).

a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794284... and c = 2.719099850893334482... - Vaclav Kotesovec, Feb 07 2018

A245768 G.f. satisfies: A(x) = 1 + xA(x)^4 / (A(x) - xA'(x)).

Original entry on oeis.org

1, 1, 4, 26, 224, 2337, 28088, 377144, 5544824, 88039724, 1494960308, 26954440490, 513267546824, 10279486681982, 215822203235952, 4737785187211908, 108509135362455192, 2588049036893027820, 64180886929824389840, 1652564046132761428040, 44124859215715377422552, 1220338620776444854394561

Offset: 0

Paul D. Hanna, Aug 01 2014

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 224*x^4 + 2337*x^5 + 28088*x^6 +...
The table of coefficients of x^k in A(x)^n begin:
n=1: [1,  1,  4,  26,  224,  2337,  28088,   377144, ...];
n=2: [1,  2,  9,  60,  516,  5330,  63318,   840808, ...];
n=3: [1,  3, 15, 103,  888,  9105, 107050,  1406655, ...];
n=4: [1,  4, 22, 156, 1353, 13804, 160844,  2092748, ...];
n=5: [1,  5, 30, 220, 1925, 19586, 226480,  2919840, ...];
n=6: [1,  6, 39, 296, 2619, 26628, 305979,  3911688, ...];
n=7: [1,  7, 49, 385, 3451, 35126, 401625,  5095392, ...];
n=8: [1,  8, 60, 488, 4438, 45296, 515988,  6501760, ...];
n=9: [1,  9, 72, 606, 5598, 57375, 651948,  8165700, ...];
n=10:[1, 10, 85, 740, 6950, 71622, 812720, 10126640, ...]; ...
in which the diagonals illustrate the relation
[x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+3) for n>=1
as follows:
[x^1] A(x)^2 = 2 = 2*[x^0] A(x)^4 = 2*1 ;
[x^2] A(x)^3 = 15 = 3*[x^1] A(x)^5 = 3*5 ;
[x^3] A(x)^4 = 156 = 4*[x^2] A(x)^6 = 4*39 ;
[x^4] A(x)^5 = 1925 = 5*[x^3] A(x)^7 = 5*385 ;
[x^5] A(x)^6 = 26628 = 6*[x^4] A(x)^8 = 6*4438 ;
[x^6] A(x)^7 = 401625 = 7*[x^5] A(x)^9 = 7*57375 ;
[x^7] A(x)^8 = 6501760 = 8*[x^6] A(x)^10 = 8*812720 ; ...
Also, from the above table, we can generate:
[1/1, 2/2, 15/3, 156/4, 1925/5, 26628/6, 401625/7, 812720/8, ...]
= [1, 1, 5, 39, 385, 4438, 57375, 812720, 12428977, 203183595, ...];
the g.f. G(x) of which begins:
G(x) = x + x^2 + 5*x^3 + 39*x^4 + 385*x^5 + 4438*x^6 + 57375*x^7 +...
such that:
G(x) = x*G(x)^4 + x^2*G(x)^3*G'(x) and G(x) = A(G(x)).

Cf. A088715, A245118.

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

/* From A(x) = 1 +  x*A(x)^4 / (A(x) - x*A'(x)): */
{a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^4/(A - x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

/* From A(x) = x/Series_Reversion(G) where G = x*G^4 + x^2*G^3*G': */
{a(n)=local(G=1+x); for(i=1, n, G = 1 + x*G^4 + x^2*G^3*G' +x*O(x^n)); polcoeff(x/serreverse(x*G +x^2*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+3) for n>=1.
(2) A(x) = x/Series_Reversion(G(x)) where G(x) = x*G(x)^4 + x^2*G(x)^3*G'(x).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A238223 Decimal expansion of a constant related to A088716.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A352236 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 2*x*A'(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A352235 G.f. A(x) satisfies: A(x) = 1 + x*A(x) / (A(x) - 3*x*A'(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A352237 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 3*x*A'(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A352238 G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 / (A(x) - 4*x*A'(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A158884 G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A295766 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2)/n^2 = [x^(n-2)] A(x)^(n^2) for n>=2 with A'(0) = 1.

Views

Author

Keywords

Comments

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

A352236 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 2x*A'(x)).

A352235 G.f. A(x) satisfies: A(x) = 1 + xA(x) / (A(x) - 3x*A'(x)).

A352237 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 3x*A'(x)).

A352238 G.f. A(x) satisfies: A(x) = 1 + xA(x)^2 / (A(x) - 4x*A'(x)).

A158884 G.f. A(x) satisfies: d/dx xA(x) = 1+x + x[d/dx log(A(x))].

A295811 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = 2n [x^(n-2)] A(x)^(n^2) for n>=2, with A(0) = 1.

A245768 G.f. satisfies: A(x) = 1 + xA(x)^4 / (A(x) - xA'(x)).