A156326 -id:A156326 - cp's OEIS frontend

A182962 E.g.f. satisfies: A(x) = exp( x/(1 - x*A'(x)/A(x)) ).

1, 1, 3, 25, 433, 12501, 529531, 30495613, 2272643745, 211761416233, 24055076979091, 3267213865097601, 522451410607362193, 97120159467079471165, 20765771676360919883403, 5060640084128464622069221

Offset: 0

Paul D. Hanna, Jan 01 2011

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 433*x^4/4! +...
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 +...+ A088716(n)*x^(n+1) +...
...
The coefficients of [x^n/n!] in the powers of e.g.f. A(x) begin:
A^1: [(1),(1), 3, 25, 433, 12501, 529531, 30495613, ...];
A^2: [1,(2),(8), 68, 1120, 30832, 1260544, 70737536, ...];
A^3: [1, 3,(15),(135), 2169, 57303, 2261439, 123523515, ...];
A^4: [1, 4, 24,(232),(3712), 94944, 3622336, 192461056, ...];
A^5: [1, 5, 35, 365, (5905),(147625), 5460475, 282185825, ...];
A^6: [1, 6, 48, 540, 8928, (220176),(7926336), 398625408, ...];
A^7: [1, 7, 63, 763, 12985, 318507,(11210479),(549313471), ...];
A^8: [1, 8, 80, 1040, 18304, 449728, 15551104,(743759360), ...];
...
In the above table, the coefficients in parenthesis are related by:
1*1 = 1; 8 = 2^2*2; 135 = 3^2*15; 3712 = 4^2*232; 147625 = 5^2*5905;
this illustrates: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n.
...
Also note that the main diagonal in the above table begins:
[1*1, 2*1, 3*5, 4*58, 5*1181, 6*36696, 7*1601497, 8*92969920, ...];
this illustrates: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n).
...
Let G(x) denote the e.g.f. of A156326:
G(x) = 1 + x + 5*x^2/2! + 58*x^3/3! + 1181*x^4/4! + 36696*x^5/5! +...
then G(x) satisfies: G(x) = A(x*G(x)) and A(x) = G(x/A(x)) where
G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A088716, A300735, A300986, A300988, A300990, A300992.

Cf. A156326, A238223.

m = 16; A[_] = 1;
Do[A[x_] = Exp[x/(1 - x A'[x]/A[x])] + O[x]^m, {m}];
CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Oct 29 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(x/(1 - x*deriv(A)/A+x*O(x^n))));n!*polcoeff(A,n)}

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

E.g.f.: A(x) = exp(x*F(x)) where F(x) = 1 + x*F(x)*d/dx[x*F(x)] is the o.g.f. of A088716.
E.g.f. satisfies: [x^n/n!] A(x)^n = n^2*[x^(n-1)/(n-1)!] A(x)^n for n>=1.
E.g.f. satisfies: [x^n/n!] A(x)^(n+1) = (n+1)*A156326(n) for n>=0.
E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where A(x*G(x)) = G(x) is the e.g.f. of A156326, which satisfies:
. G(x) = exp( Sum_{n>=1} n^2 * A156326(n-1)*x^n/n! ).
a(n) ~ c * (n!)^2 * n, where c = 0.21795078944715106549... (see A238223). - Vaclav Kotesovec, Feb 22 2014

A143925 E.g.f. A(x) satisfies A(x) = exp(x + x^2*A'(x)).

Original entry on oeis.org

1, 1, 3, 25, 397, 10061, 369061, 18415825, 1197307161, 98248658905, 9928361978281, 1211474323983221, 175635827999270629, 29845580180227776277, 5876070628821158239293, 1327055145216772464211321, 340793190982323564066166321, 98752652958563191504390390577

Offset: 0

Paul D. Hanna, Sep 06 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A386505, A386506, A386507, A386508, A386509.
Cf. A386510, A386511, A386512, A386513.
Cf. A156326.

{a(n)=local(A=1+x);for(i=0,n,A=exp(x+x^2*deriv(A)+x*O(x^n)));n!*polcoeff(A,n)}

a(n) ~ c * (n!)^2, where c = 0.80799863989712622039... - Vaclav Kotesovec, Feb 22 2014

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k * binomial(n-1,k) * a(k) * a(n-1-k). - Seiichi Manyama, Jul 24 2025

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).

A156325 E.g.f.: A(x) = exp( Sum_{n>=1} n(n+1)/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, 4, 34, 482, 10056, 286372, 10591372, 491169996, 27826318000, 1887581200256, 150885500428224, 14028718134958936, 1500672248541122944, 182987661921689610000, 25231215606822797450176, 3906382859414378995123088, 674864208557744341737593088, 129369340822722468679389762496

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 482*x^4/4! + 10056*x^5/5! +...
log(A(x)) = x + 3*1*x^2/2! + 6*4*x^3/3! + 10*34*x^4/4! + 15*482*x^5/5! +...
such that log(A(x)) = x*A(x) + x^2*A'(x)/2 = d/dx x^2*A(x)/2.

Cf. A156326, A156327.

Cf. A385101, A386533, A386534.

terms = 19; A[] = 1; Do[A[x] = Exp[x*A[x]+x^2*A'[x]/2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)

{a(n) = if(n==0,1,n!*polcoeff(exp(sum(k=1,n,k*(k+1)/2*a(k-1)*x^k/k!)+x*O(x^n)),n))}
for(n=0,25,print1(a(n),", "))

{a(n) = if(n==0,1,sum(k=1,n,k*(k+1)/2*binomial(n-1,k-1)*a(k-1)*a(n-k)))}
for(n=0,25,print1(a(n),", "))

{a(n) = my(A=1); for(i=1,n, A = exp(deriv(x^2*A/2 +x^2*O(x^n)))); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Dec 17 2017

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+j/2)*binomial(i-1, j)*v[j+1]*v[i-j])); v; \\ Seiichi Manyama, Jul 25 2025

a(n) = Sum_{k=1..n} k(k+1)/2 * C(n-1,k-1)*a(k-1)*a(n-k) for n>0, with a(0)=1.
E.g.f. A(x) satisfies A(x) = exp( d/dx x^2*A(x)/2 ). - Paul D. Hanna, Dec 17 2017
a(n) ~ c * n!^2 * n^3 / 2^n, where c = 0.1298868467763756870740708... - Vaclav Kotesovec, Aug 05 2025

A385920 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^3A''(x)).

Original entry on oeis.org

1, 1, 3, 34, 1085, 76176, 10075567, 2259237184, 795650626521, 415436957516800, 307467426910853051, 311183690415601457664, 418253671031607891057877, 728624453608629352377831424, 1611758187912750506708147828775, 4448533739124778044473142239512576

Offset: 0

Seiichi Manyama, Jul 12 2025

nonn

Cf. A000272, A156326, A385921, A385922, A385923.

Cf. A385762.

terms = 16; A[] = 1; Do[A[x] = Exp[x*A[x]+x^3*A''[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 2, stirling(2, k, 1)*j^k))*binomial(i-1, j)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

A385921 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^4A'''(x)).

Original entry on oeis.org

1, 1, 3, 16, 509, 66216, 24639367, 21043463344, 35690424280569, 108571039785256960, 549371080081204026731, 4363111116508031602712064, 51938511093491129409954627637, 892615592639462586040781503568896, 21469194967164193484102627607895188975, 703974996795045871424921458192403079479296

Offset: 0

Seiichi Manyama, Jul 12 2025

nonn

Cf. A000272, A156326, A385920, A385922, A385923.

Cf. A385763.

terms = 16; A[] = 1; Do[A[x] = Exp[x*A[x]+x^4*A'''[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 3, stirling(3, k, 1)*j^k))*binomial(i-1, j)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + 2*k - 3*k^2 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

A385922 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^5A''''(x)).

Original entry on oeis.org

1, 1, 3, 16, 125, 16296, 11929927, 30230776864, 203634850471929, 3082625458810336000, 93280255561776693446891, 5173509703646410927969711104, 491814532626655136406839912703157, 75968624000349445912469318939348786176, 18252829396078618393615717880609268502659375

Offset: 0

Seiichi Manyama, Jul 12 2025

nonn

Cf. A000272, A156326, A385920, A385921, A385923.

Cf. A385764.

terms = 15; A[] = 1; Do[A[x] = Exp[x*A[x]+x^5*A''''[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 4, stirling(4, k, 1)*j^k))*binomial(i-1, j)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 - 6*k + 11*k^2 - 6*k^3 + k^4) * binomial(n-1,k) * a(k) * a(n-1-k).

A385939 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^2) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 88, 3893, 352536, 57322537, 15277686880, 6239711818377, 3708478187297920, 3079046917046731661, 3455392385954013825024, 5100835934217411940938685, 9682263835381845999967986688, 23180826149963609282826172967025, 68850271609123855250628849758027776

Offset: 0

Seiichi Manyama, Jul 13 2025

nonn

Cf. A156326, A385940, A385941, A385942, A385943.

Cf. A385830.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+j^2)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

E.g.f. A(x) satisfies A(x) = exp( x * A(x) + x^2 * (d/dx A(x)) + x^3 * (d^2/dx^2 A(x)) ).

A385940 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 148, 17189, 5676336, 4326290857, 6602349049360, 18222895109730537, 84299882148193513600, 616234715187848381357261, 6792153358905298302629935104, 108647409624774384033524243233165, 2443481854821246436998727854436139008, 75225062360951292682727255438183855480625

Offset: 0

Seiichi Manyama, Jul 13 2025

nonn

Cf. A156326, A385939, A385941, A385942, A385943.

Cf. A385831.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+j^3)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

E.g.f. A(x) satisfies A(x) = exp( x*A(x) + x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ).

A385941 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^4) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 268, 88997, 114813696, 431933720137, 3924557764490560, 75445736579647162857, 2782590090487142758353280, 182621397948270167786531824781, 20092371907364577184989521575079424, 3530551258386563793887714321816262653965, 951815440668013126114976449397609983348430848

Offset: 0

Seiichi Manyama, Jul 13 2025

nonn

Cf. A156326, A385939, A385940, A385942, A385943.

Cf. A385832.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+j^4)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

E.g.f. A(x) satisfies A(x) = exp( x*A(x) + x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ).

cp's OEIS Frontend

A182962 E.g.f. satisfies: A(x) = exp( x/(1 - x*A'(x)/A(x)) ).

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A143925 E.g.f. A(x) satisfies A(x) = exp(x + x^2*A'(x)).

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A238223 Decimal expansion of a constant related to A088716.

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A156325 E.g.f.: A(x) = exp( Sum_{n>=1} n(n+1)/2 * a(n-1)*x^n/n! ) = Sum_{n>=0} a(n)*x^n/n! with a(0)=1.

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A385920 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^3*A''(x)).

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A385921 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^4*A'''(x)).

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A385922 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^5*A''''(x)).

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A385939 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^2) * binomial(n-1,k) * a(k) * a(n-1-k).

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A385940 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

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A156325 E.g.f.: A(x) = exp( Sum_{n>=1} n(n+1)/2 * a(n-1)x^n/n! ) = Sum_{n>=0} a(n)x^n/n! with a(0)=1.

A385920 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^3A''(x)).

A385921 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^4A'''(x)).

A385922 E.g.f. A(x) satisfies A(x) = exp(xA(x) + x^5A''''(x)).