A326433 -id:A326433 - cp's OEIS frontend

A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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, 682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147, 51724158235372, 101557600812015, 82635818516305, 36672690416280, 9831937482310, 1665456655065, 180791918475, 12443391060, 520878315, 12004575, 115975

Offset: 0

Paul D. Hanna, Jul 20 2019

			E.g.f.: A(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! + (682076806159 + 1267153412532*y + 959061013824*y^2 + 387848415927*y^3 + 92381300277*y^4 + 13455280629*y^5 + 1200540180*y^6 + 63424134*y^7 + 1805067*y^8 + 21147*y^9)*x^9/9! + (51724158235372 + 101557600812015*y + 82635818516305*y^2 + 36672690416280*y^3 + 9831937482310*y^4 + 1665456655065*y^5 + 180791918475*y^6 + 12443391060*y^7 + 520878315*y^8 + 12004575*y^9 + 115975*y^10)*x^10/10! + ...
such that
A(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
A(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! + ...).
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(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],
[682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147], ...
Main diagonal is A000110 (Bell numbers).
Leftmost column is A020557(n) = A000110(2*n), for n >= 0.
Row sums form A326433.

Paul D. Hanna, Table of n, a(n) for n = 0..495 (first 30 rows of this triangle).

Cf. A000110, A020557, A108459, A326433, A326434, A326435, A326436, A326437.

Cf. A326601 (central terms).

E.g.f.: exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!.
E.g.f.: exp(-1-y) * Sum_{n>=0} exp(n^2*x) * exp( y*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
T(n,n) = A000110(n) for n >= 0, where A000110 is the Bell numbers.
T(n,0) = A000110(2*n) for n >= 0, where A000110 is the Bell numbers.
Sum_{k=0..n} T(n,k) * (-1)^k = A108459(n) for n >= 0.
Sum_{k=0..n} T(n,k) = A326433(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 2^k = A326434(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 3^k = A326435(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 4^k = A326436(n) 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!.

A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.

Original entry on oeis.org

1, 3, 22, 297, 6055, 169431, 6145827, 277912452, 15225719420, 988814989679, 74822364609113, 6505084496930641, 642317112612827029, 71331999557857791694, 8835651007377368848464, 1211946040741011512724559, 182930472229597183037431011, 30216143201862939999461382959, 5435054718681965118312689633935

Offset: 0

Paul D. Hanna, Jul 09 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 = x, r = 1.

			E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 297*x^3/3! + 6055*x^4/4! + 169431*x^5/5! + 6145827*x^6/6! + 277912452*x^7/7! + 15225719420*x^8/8! + 988814989679*x^9/9! + 74822364609113*x^10/10! + ...
such that
A(x) = exp(-1) * (1 + (exp(x) + x) + (exp(2*x) + x)^2/2! + (exp(3*x) + x)^3/3! + (exp(4*x) + x)^4/4! + (exp(5*x) + x)^5/5! + (exp(6*x) + x)^6/6! + (exp(7*x) + x)^7/7! + (exp(8*x) + x)^8/8! + ...)
also,
A(x) = exp(-1) * (exp(x) + exp(x)*exp(x*exp(x)) + exp(4*x)*exp(x*exp(2*x))/2! + exp(9*x)*exp(x*exp(3*x))/3! + exp(16*x)*exp(x*exp(4*x))/4! + exp(25*x)*exp(x*exp(5*x))/5! + exp(36*x)*exp(x*exp(6*x))/6! + ...).

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

Cf. A326433.

/* Requires appropriate precision */
\p200
{a(n) = my(A = exp(-1) * sum(m=0,n+300, (exp(m*x +x*O(x^n)) + x)^m / m! )); round(n!*polcoeff(A,n))}
for(n=0,20,print1(a(n),", "))

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

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

A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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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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A326430 E.g.f.: exp(-1) * Sum_{n>=0} (exp(n*x) + x)^n / n!.

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

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A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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