A326600 -id:A326600 - cp's OEIS frontend

A326601 Central terms in triangle A326600.

1, 12, 1947, 945360, 952279230, 1665456655065, 4546140251381410, 18036839485026245312, 98828821396412329832181, 719565439198091448998634599, 6755047194370225050422094037182, 79804922388776170830478631955052404, 1163101849742231572210960509481022794738, 20565795904976685513209147957073892094206920, 434964979224032851486461932786665860631628717100, 10870355119499979196080422944546745278774481226306000

Offset: 0

Paul D. Hanna, Jul 22 2019

nonn

			E.g.f. of A326600 begins
F(x,y) = 1 + (2 + y)*x + (15 + 12*y + 2*y^2)*x^2/2! + (203 + 206*y + 60*y^2 + 5*y^3)*x^3/3! + (4140 + 4949*y + 1947*y^2 + 298*y^3 + 15*y^4)*x^4/4! + (115975 + 156972*y + 75595*y^2 + 16160*y^3 + 1535*y^4 + 52*y^5)*x^5/5! + (4213597 + 6301550*y + 3528368*y^2 + 945360*y^3 + 127915*y^4 + 8307*y^5 + 203*y^6)*x^6/6! + (190899322 + 310279615*y + 195764198*y^2 + 62079052*y^3 + 10690645*y^4 + 1001567*y^5 + 47397*y^6 + 877*y^7)*x^7/7! + (10480142147 + 18293310174*y + 12735957930*y^2 + 4614975428*y^3 + 952279230*y^4 + 114741060*y^5 + 7901236*y^6 + 285096*y^7 + 4140*y^8)*x^8/8! + ...
such that
F(x,y) = exp(-1-y) * (1 + (exp(x) + y) + (exp(2*x) + y)^2/2! + (exp(3*x) + y)^3/3! + (exp(4*x) + y)^4/4! + (exp(5*x) + y)^5/5! + (exp(6*x) + y)^6/6! + ...)
also
F(x,y) = exp(-1-y) * (exp(y) + exp(x)*exp(y*exp(x)) + exp(4*x)*exp(y*exp(2*x))/2! + exp(9*x)*exp(y*exp(3*x))/3! + exp(16*x)*exp(y*exp(4*x))/4! + exp(25*x)*exp(y*exp(5*x))/5! + exp(36*x)*exp(y*exp(6*x))/6! + ...).
Triangle A326600 of coefficients of x^n*y^k/n! in F(x,y) begins:
[1],
[2, 1],
[15, 12, 2],
[203, 206, 60, 5],
[4140, 4949, 1947, 298, 15],
[115975, 156972, 75595, 16160, 1535, 52],
[4213597, 6301550, 3528368, 945360, 127915, 8307, 203],
[190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877],
[10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140], ...
in which the central terms form this sequence.

Cf. A326600.

a(n) = [x^(2*n)*y^n/n!] exp(-1-y) * Sum_{m>=0} (exp(m*x) + y)^m / m!.

a(n) = [x^(2*n)*y^n/n!] exp(-1-y) * Sum_{m>=0} exp(m^2*x) * exp( y*exp(m*x) ) / m!.

A108459 Number of labeled partitions of (n,n) into pairs (i,j).

Original entry on oeis.org

1, 1, 5, 52, 855, 19921, 614866, 24040451, 1152972925, 66200911138, 4465023867757, 348383154017581, 31052765897026352, 3128792250765898965, 353179564583216567917, 44320731930172534543092, 6141797839043095806714667, 934330605640859569909566925

Offset: 0

Christian G. Bower, Jun 03 2005

nonn

Partitions of n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one white object.

a(n) is also the number of elements of the partition monoid P_n with domain {1,...,n}. Elements of P_n are set partitions of {1,1',...,n,n'}, and the domain of such a partition is the set of all points in {1,...,n} that belong to a block containing a dashed element. - James East, Apr 10 2018

Seiichi Manyama, Table of n, a(n) for n = 0..200

Main diagonal of A108458. Cf. A108461.
Cf. A048993 (Stirling2), A068424 (falling factorial).
Cf. A326600, A326270, A326271, A326288.
Bisection of A124421 (even part).

b:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
a:= n-> add(coeff(b(n), x, j)*j^n, j=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 02 2023

a[n_] := If[n == 0, 1, Sum[k^n*StirlingS2[n, k], {k, 0, n}]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2024 *)

{a(n)=polcoeff(sum(m=0, n, m^m*x^m/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)} \\ Paul D. Hanna, Sep 17 2013

{a(n)=n!*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!), n)} \\ Paul D. Hanna, Sep 17 2013

a(n) = Sum_{k=0..n} k^n*Stirling2(n,k). - Vladeta Jovovic, Aug 31 2006
E.g.f.: Sum_{n>=0} (exp(n*x)-1)^n / n!. - Vladeta Jovovic, Jul 12 2007
E.g.f.: Sum_{n>=0} exp(n^2*x) * exp( -exp(n*x) ) / n!. - Paul D. Hanna, Jun 28 2019
O.g.f.: Sum_{n>=0} n^n * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Sep 17 2013
a(n) = Sum_{k=0..n} Stirling2(n,k) * Sum_{l=k..n} Stirling2(n,l)*T(l,k). Here T(l,k) are the falling factorials. - James East, Apr 10 2018

A326433 E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

Original entry on oeis.org

1, 3, 29, 474, 11349, 366289, 15125300, 770762673, 47199596441, 3403242019876, 284281430425747, 27150503912943937, 2932403885598294838, 354869660881411722107, 47739034071736749352125, 7090201955561116768761250, 1155624866838027573814278801, 205611555585528308269669174557, 39746979329229607204823274477284

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 1, r = 1.

			E.g.f.: A(x) = 1 + 3*x + 29*x^2/2! + 474*x^3/3! + 11349*x^4/4! + 366289*x^5/5! + 15125300*x^6/6! + 770762673*x^7/7! + 47199596441*x^8/8! + 3403242019876*x^9/9! + 284281430425747*x^10/10! + 27150503912943937*x^11/11! + 2932403885598294838*x^12/12! + ...
such that
A(x) = exp(-2) * (1 + (exp(x) + 1) + (exp(2*x) + 1)^2/2! + (exp(3*x) + 1)^3/3! + (exp(4*x) + 1)^4/4! + (exp(5*x) + 1)^5/5! + (exp(6*x) + 1)^6/6! + ...)
also
A(x) = exp(-2) * (exp(1) + exp(x)*exp(exp(x)) + exp(4*x)*exp(exp(2*x))/2! + exp(9*x)*exp(exp(3*x))/3! + exp(16*x)*exp(exp(4*x))/4! + exp(25*x)*exp(exp(5*x))/5! + exp(36*x)*exp(exp(6*x))/6! + ...).

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

Cf. A326600, A020557, A326430, A326434, A326435, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-2) * sum(n=0,500, (exp(n*x +O(x^31)) + 1)^n/n! ) )))

E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.
E.g.f.: exp(-2) * Sum_{n>=0} exp(n^2*x) * exp( exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

Original entry on oeis.org

1, 4, 47, 895, 24450, 887803, 40818505, 2297393888, 154381810471, 12149510583583, 1102672816721422, 113974516318639363, 13277046519634998953, 1727765194711759098324, 249264545884060054668295, 39606622952407779396832791, 6891271396238954765341535650, 1306288225868329080524305347859, 268542657134280438710389415260401, 59628381166607045580114829853101712

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 2, r = 1.

			E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 895*x^3/3! + 24450*x^4/4! + 887803*x^5/5! + 40818505*x^6/6! + 2297393888*x^7/7! + 154381810471*x^8/8! + 12149510583583*x^9/9! + 1102672816721422*x^10/10! + ...
such that
A(x) = exp(-3) * (1 + (exp(x) + 2) + (exp(2*x) + 2)^2/2! + (exp(3*x) + 2)^3/3! + (exp(4*x) + 2)^4/4! + (exp(5*x) + 2)^5/5! + (exp(6*x) + 2)^6/6! + ...)
also
A(x) = exp(-3) * (exp(2) + exp(x)*exp(2*exp(x)) + exp(4*x)*exp(2*exp(2*x))/2! + exp(9*x)*exp(2*exp(3*x))/3! + exp(16*x)*exp(2*exp(4*x))/4! + exp(25*x)*exp(2*exp(5*x))/5! + exp(36*x)*exp(2*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326435, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-3) * sum(n=0, 500, (exp(n*x +O(x^31)) + 2)^n/n! ) )))

E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.
E.g.f.: exp(-3) * Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

Original entry on oeis.org

1, 5, 69, 1496, 45771, 1840537, 92925982, 5705543791, 416015394341, 35365673566750, 3454046493504337, 382930667897753421, 47708365129614794580, 6622948820406278058625, 1016977626656613380728781, 171637260767262574245781800, 31661205827344145981298200207, 6352045190999137085697971335893

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 3, r = 1.

			E.g.f.: A(x) = 1 + 5*x + 69*x^2/2! + 1496*x^3/3! + 45771*x^4/4! + 1840537*x^5/5! + 92925982*x^6/6! + 5705543791*x^7/7! + 416015394341*x^8/8! + 35365673566750*x^9/9! + 3454046493504337*x^10/10! + ...
such that
A(x) = exp(-4) * (1 + (exp(x) + 3) + (exp(2*x) + 3)^2/2! + (exp(3*x) + 3)^3/3! + (exp(4*x) + 3)^4/4! + (exp(5*x) + 3)^5/5! + (exp(6*x) + 3)^6/6! + ...)
also
A(x) = exp(-4) * (exp(3) + exp(x)*exp(3*exp(x)) + exp(4*x)*exp(3*exp(2*x))/2! + exp(9*x)*exp(3*exp(3*x))/3! + exp(16*x)*exp(3*exp(4*x))/4! + exp(25*x)*exp(3*exp(5*x))/5! + exp(36*x)*exp(3*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326434, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-4) * sum(n=0, 500, (exp(n*x +O(x^31)) + 3)^n/n! ) )))

E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
E.g.f.: exp(-4) * Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

A326436 E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

Original entry on oeis.org

1, 6, 95, 2307, 78000, 3433831, 188460821, 12508220886, 981371259995, 89426179550623, 9331384489007032, 1102143627943740931, 145924317814992561097, 21480095845779426077750, 3490477008130417972086807, 622292123277813938275834747, 121062971468108753273621477712, 25577093024015935514169919403295

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 4, r = 1.

			E.g.f.: A(x) = 1 + 6*x + 95*x^2/2! + 2307*x^3/3! + 78000*x^4/4! + 3433831*x^5/5! + 188460821*x^6/6! + 12508220886*x^7/7! + 981371259995*x^8/8! + 89426179550623*x^9/9! + 9331384489007032*x^10/10! + ...
such that
A(x) = exp(-5) * (1 + (exp(x) + 4) + (exp(2*x) + 4)^2/2! + (exp(3*x) + 4)^3/3! + (exp(4*x) + 4)^4/4! + (exp(5*x) + 4)^5/5! + (exp(6*x) + 4)^6/6! + ...)
also
A(x) = exp(-5) * (exp(4) + exp(x)*exp(4*exp(x)) + exp(4*x)*exp(4*exp(2*x))/2! + exp(9*x)*exp(4*exp(3*x))/3! + exp(16*x)*exp(4*exp(4*x))/4! + exp(25*x)*exp(4*exp(5*x))/5! + exp(36*x)*exp(4*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326434, A326435, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (exp(n*x +O(x^31)) + 4)^n/n! ) )))

E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.
E.g.f.: exp(-5) * Sum_{n>=0} exp(n^2*x) * exp( 4*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

A326437 E.g.f.: exp(-5) * Sum_{n>=0} (2exp(nx) + 3)^n / n!.

Original entry on oeis.org

1, 12, 298, 11154, 568004, 37059182, 2978383982, 286712714932, 32370944416718, 4216616929161674, 625354679867770896, 104450484419292872298, 19469192354728354857686, 4018460441266469063161936, 912287005016859245973405858, 226476227666270561445555706042, 61164205107875867322971316940164

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 3/2, r = 2.

			E.g.f.: A(x) = 1 + 12*x + 298*x^2/2! + 11154*x^3/3! + 568004*x^4/4! + 37059182*x^5/5! + 2978383982*x^6/6! + 286712714932*x^7/7! + 32370944416718*x^8/8! + 4216616929161674*x^9/9! + ...
such that
A(x) = exp(-5) * (1 + (2*exp(x) + 3) + (2*exp(2*x) + 3)^2/2! + (2*exp(3*x) + 3)^3/3! + (2*exp(4*x) + 3)^4/4! + (2*exp(5*x) + 3)^5/5! + (2*exp(6*x) + 3)^6/6! + ...)
also
A(x) = exp(-5) * (exp(3) + 2*exp(x)*exp(3*exp(x)) + 2^2*exp(4*x)*exp(3*exp(2*x))/2! + 2^3*exp(9*x)*exp(3*exp(3*x))/3! + 2^4*exp(16*x)*exp(3*exp(4*x))/4! + 2^5*exp(25*x)*exp(3*exp(5*x))/5! + 2^6*exp(36*x)*exp(3*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326434, A326435, A326436.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (2*exp(n*x +O(x^31)) + 3)^n/n! ) )))

E.g.f.: exp(-5) * Sum_{n>=0} (2*exp(n*x) + 3)^n / n!.

E.g.f.: exp(-5) * Sum_{n>=0} 2^n * exp(n^2*x) * exp( 3*exp(n*x) ) / n!.

A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2Pii)^n/n! = C + i*S such that C^2 + S^2 = 1.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Apr 09 2024

Related identity: Sum_{n>=0} (x^n + y)^n/n! = Sum_{n>=0} exp(y*x^n)*x^(n^2)/n!. Here, x = r and y = 2*Pi*i.

What are the roots of Norm( Sum_{n>=0} (x^n + 2*Pi*i)^n/n! ) = 1? The real roots include x = 0 and x = r (this constant).

			The initial 500 digits of this constant r are
r = 0.27343900190856838553879175800469815024017455601953\
74037957877464509350868878428665975433874229621952\
21271807208862504474781327669150216691806622917186\
30052292342530146845288659570856888661537928135397\
91914154858221560663972999347727219299210079054658\
20785838554943078876634169703813817526574697076018\
43103025671330263969269247113168608393647224573552\
82695245129846145197371729802801821910764770241403\
85315562772171090016733480930506290614196661276630\
35680469795753191100711562687066719873558759501438...
Given Sum_{n>=0} (r^n + 2*Pi*i)^n / n! = C + i*S
then C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!, where
C = 0.96236940120128609855708390989630224707797733780139\
33346689286186097367092064030604732865267035268595\
44279783779811281344593178122348416729686502694192\
27215955652725928674242226419071059523037649451781\
36060669147586159699815697962817267659814744582224\
93126268783872251860132042094952557434607056861286\
20902477149931860926346847824008347947488598827305\
47837372109484356517193566333052743194953698066525\
72228584587713864226102674129509160583381421007047\
75118828482389128699072732009353421657729660481717...
and S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!, where
S = 0.27174461472396842102515050866607715426951746748919\
04159412993022348271493896385066506863889535797824\
35085649751784233166430963459007191963331589808443\
52259856849111637575812332490848107413710402589323\
75221334357855133874979455560441735994213395179878\
38917993730963815574520261440791182088848636006332\
68221934823032560291871222621378256174374612116671\
09358271083370500808439006024716239994653435216572\
21204963868973568338610259219318795040671357965613\
68248089245008828798740589773672045329008665505374...
such that C^2 + S^2 = 1.

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A326600.

Constant r and related values C and S satisfy the following formulas.
(1) Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.
(2) C = Sum_{n>=0} cos(2*Pi*r^n) * r^(n^2) / n!.
(3) S = Sum_{n>=0} sin(2*Pi*r^n) * r^(n^2) / n!.

A371710 Expansion of e.g.f. A(x) satisfying Sum_{n>=0} (A(x)^n - x)^n / n! = 1.

Original entry on oeis.org

1, -1, 7, -37, 371, -4741, 72885, -1380541, 29678815, -726074821, 19834534193, -597434105005, 19716033256947, -706675332962509, 27351721308658141, -1136955116183424829, 50513209770352997927, -2388911790071698253845, 119817073596530701766985, -6352554087532686682163053

Offset: 1

Paul D. Hanna, Apr 10 2024

sign

Related identity: Sum_{n>=0} (q^n + p)^n * r^n/n! = Sum_{n>=0} exp(p*q^n*r) * q^(n^2) * r^n/n!; here, q = A(x), p = -x, r = 1.

Conjecture: for n > 0, a(6*n + k) == [2,0,2,1,2,2] (mod 3) at k = [0,1,2,3,4,5], respectively.

			E.g.f.: A(x) = x - x^2/2! + 7*x^3/3! - 37*x^4/4! + 371*x^5/5! - 4741*x^6/6! + 72885*x^7/7! - 1380541*x^8/8! + 29678815*x^9/9! - 726074821*x^10/10! + ...
where e.g.f. A(x) satisfies the following sums.
(1) 1 = 1 + (A(x) - x) + (A(x)^2 - x)^2/2! + (A(x)^3 - x)^3/3! + (A(x)^4 - x)^4/4! + (A(x)^5 - x)^5/5! + ...
(2) 1 = exp(-x) + exp(-x*A(x))*A(x) + exp(-x*A(x)^2)*A(x)^4/2! + exp(-x*A(x)^3)*A(x)^9/3! + exp(-x*A(x)^4)*A(x)^16/4! + ...
SPECIFIC VALUES.
A(1/3) = 0.309336999832107073180903710282149168034207161078640395207...
A(1/4) = 0.232922937634173409470673241764259081533730452334005659588...
A(1/5) = 0.187560199855301209894398645611115284037479048219241021351...
A(-1/3) = -0.5146620783815103062311605400508155869729182062358910349...
A(-1/4) = -0.3123628005245983090140211998639545568283470783996606926...
A(-1/5) = -0.2335665203884038676850050992335539648367581317265287642...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A326600, A326097, A326433.

/* Sum_{n>=0} exp(-x*A(x)^n) * A(x)^(n^2) / n! = 1 */
{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( sum(m=0,sqrtint(#A+1), exp(-x*Ser(A)^m +x*O(x^#A)) * Ser(A)^(m^2)/m! ), #A-1); ); n!*A[n+1]}
for(n=1,30,print1(a(n),", "))

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

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) Sum_{n>=0} (A(x)^n - x)^n / n! = 1.
(2) Sum_{n>=0} exp(-x*A(x)^n) * A(x)^(n^2) / n! = 1.
a(n) ~ -c * n^(n-1) / (exp(n) * r^n), where c = sqrt( (-r*Sum_{k>=0} ((k*(-r + s^k)^(-1 + k))/k!)) / Sum_{k>=0} ((-1 + k)*k^2 * s^(-2 + k)*(-r + s^k)^(-2 + k)*(-r + (1 + k)*s^k))/k! ) = 0.4187940561612508319941857365856199965995537726..., and where r = -0.3491010753747466229482161022113556770139942642631... and s = -0.6426571454882319173136663034041471668334385049965... are roots of the equations Sum_{k>=0} (-r + s^k)^k/k! = 1 and Sum_{k>=0} (k^2*s^(-1 + k)*(-r + s^k)^(-1 + k))/k! = 0. - Vaclav Kotesovec, Apr 11 2024
The values r and A(r) = s given above also satisfy Sum_{n>=0} n*(n - r*A(r)^n) * exp(-r*A(r)^n) * A(r)^(n^2) / n! = 0. - Paul D. Hanna, Apr 12 2024

cp's OEIS Frontend

A326601 Central terms in triangle A326600.

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A108459 Number of labeled partitions of (n,n) into pairs (i,j).

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A326433 E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

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A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

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A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

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A326436 E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

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A326437 E.g.f.: exp(-5) * Sum_{n>=0} (2*exp(n*x) + 3)^n / n!.

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A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2*Pi*i)^n/n! = C + i*S such that C^2 + S^2 = 1.

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A326437 E.g.f.: exp(-5) * Sum_{n>=0} (2exp(nx) + 3)^n / n!.

A371707 Constant r > 0 satisfying: Sum_{n>=0} (r^n + 2Pii)^n/n! = C + i*S such that C^2 + S^2 = 1.