A080108 -id:A080108 - cp's OEIS frontend

A245835 E.g.f.: exp( x(2 + exp(3x)) ).

1, 3, 15, 108, 945, 9558, 109917, 1412316, 19959777, 306805482, 5087064789, 90370321704, 1710170426097, 34308056537550, 726612812416269, 16188742781216892, 378244417385086785, 9242436410233527762, 235609985190361119525, 6252379688953421699760, 172380307421633200750161

Offset: 0

Paul D. Hanna, Aug 02 2014

nonn

			E.g.f.: E(x) = 1 + 3*x + 15*x^2/2! + 108*x^3/3! + 945*x^4/4! + 9558*x^5/5! +...
where E(x) = exp(2*x) * exp(x*exp(3*x)).
O.g.f.: A(x) = 1 + 3*x + 15*x^2 + 108*x^3 + 945*x^4 + 9558*x^5 + 109917*x^6 +...
where
A(x) = 1/(1-2*x) + x/(1-5*x)^2 + x^2/(1-8*x)^3 + x^3/(1-11*x)^4 + x^4/(1-14*x)^5 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A080108, A216689, A240165, A245834.

Table[Sum[Binomial[n,k] * (3*k+2)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)
With[{nn=20},CoefficientList[Series[Exp[x(2+Exp[3x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 06 2015 *)

{a(n)=local(A=1);A=exp( x*(2 + exp(3*x +x*O(x^n))) );n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (3*k+2)*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

{a(n)=sum(k=0,n,(3*k+2)^(n-k)*binomial(n,k))}
for(n=0,30,print1(a(n),", "))

O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+2)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+2)^(n-k) for n>=0.
a(n) ~ exp((n+6*r^2)/(1+3*r)) * n! / (r^n*sqrt(2*Pi*(-6*r^2*(2+3*r) + n*(1+9*r+9*r^2)) / (1+3*r))), where r is the root of the equation r*(2 + (1+3*r)*exp(3*r)) = n. - Vaclav Kotesovec, Aug 03 2014
(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)). - Vaclav Kotesovec, Aug 06 2014

A355471 Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.

Original entry on oeis.org

1, 1, 2, 10, 77, 808, 11257, 196072, 4136897, 103755904, 3034193921, 101901347944, 3885951145969, 166605168800704, 7961498177012993, 420976047757358776, 24475992585921169553, 1556007778666449968128, 107625967130820901112833

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A080108, A135746, A234568, A355463, A355472.

Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^2*x))^k))

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

a(n) = Sum_{k=1..n} k^(2*(n-k)) * binomial(n-1,k-1) for n > 0.

A367874 Expansion of e.g.f. exp(x * (2 + exp(x))).

Original entry on oeis.org

1, 3, 11, 48, 241, 1358, 8445, 57256, 419233, 3290202, 27507349, 243731084, 2278919697, 22402234390, 230781192301, 2484462888312, 27880896280513, 325432611292082, 3943062342781605, 49504837209940612, 642982531293731761, 8626753575445207278

Offset: 0

Seiichi Manyama, Dec 03 2023

Cf. A000248, A080108, A367875.

Cf. A240165.

a(n) = sum(k=0, n, (k+2)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k / (1 - (k+2)*x)^(k+1).
a(n) = Sum_{k=0..n} (k+2)^(n-k) * binomial(n,k).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * k * a(n-k). - Ilya Gutkovskiy, Feb 02 2024

A196794 a(n) = Sum_{k=0..n} binomial(n,k)2^k(k+1)^(n-k).

Original entry on oeis.org

1, 3, 13, 69, 425, 2953, 22701, 190445, 1725777, 16757649, 173244629, 1896821941, 21897166137, 265525063001, 3371067773565, 44683137692157, 616811052816545, 8847765111928609, 131622808197394341, 2027097866771329349, 32267707989783480201, 530125689222591861993

Offset: 0

Paul D. Hanna, Oct 06 2011

nonn

Robert Israel, Table of n, a(n) for n = 0..523

Cf. A080108, A196795.

S:= series(exp(x+2*x*exp(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jan 20 2017

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

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

{a(n)=n!*polcoeff(exp(x+2*x*exp(x+x*O(x^n))),n)}

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

E.g.f.: exp(x + 2*x*exp(x)).

A116072 Central terms of triangle A116071, which equals Pascal's triangle to the matrix power of Pascal's triangle.

Original entry on oeis.org

1, 2, 18, 200, 2870, 49392, 976668, 21697104, 532727910, 14275220960, 413469332276, 12845983030608, 425442421627132, 14941814934855200, 554044899080129400, 21608731448473756320, 883563752144886420870

Offset: 0

Paul D. Hanna, Feb 03 2006

nonn

Cf. A000248, A080108, A116071.

{a(n)=local(X=x+x*O(x^(2*n)),Y=y+y*O(y^n)); (2*n)!*polcoeff(polcoeff(exp(X*(Y+exp(X))),2*n,x),n,y)}

a(n) = (n+1)*A000108(n)*A000248(n).

A196795 a(n) = Sum_{k=0..n} binomial(n,k)3^k(k+1)^(n-k).

Original entry on oeis.org

1, 4, 22, 145, 1096, 9259, 85924, 865183, 9364864, 108173827, 1325589676, 17149360111, 233271228880, 3324545097475, 49493784653644, 767665750130839, 12376226335249024, 206967901014192643, 3583561993192959436, 64136093489935863583, 1184711492540805987856

Offset: 0

Paul D. Hanna, Oct 06 2011

nonn

Cf. A080108, A196794.

Table[Sum[Binomial[n,k]3^k (k+1)^(n-k),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Nov 12 2012 *)

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

{a(n)=polcoeff(sum(m=0,n,3^m*x^m/(1-(m+1)*x+x*O(x^n))^(m+1)),n)}

{a(n)=n!*polcoeff(exp(x+3*x*exp(x+x*O(x^n))),n)}

O.g.f.: Sum_{n>=0} 3^n*x^n/(1 - (n+1)*x)^(n+1).

E.g.f.: exp(x + 3*x*exp(x)).

A367875 Expansion of e.g.f. exp(x * (3 + exp(x))).

Original entry on oeis.org

1, 4, 18, 91, 512, 3169, 21352, 155257, 1209680, 10039825, 88318136, 819958033, 8004898600, 81913041721, 876117919616, 9770201709649, 113347591376672, 1365288066794017, 17043527322085096, 220145837754233713, 2937871757773069496, 40451715334029650953

Offset: 0

Seiichi Manyama, Dec 03 2023

Cf. A000248, A080108, A367874.

Cf. A245834.

a(n) = sum(k=0, n, (k+3)^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} x^k / (1 - (k+3)*x)^(k+1).
a(n) = Sum_{k=0..n} (k+3)^(n-k) * binomial(n,k).
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * k * a(n-k). - Ilya Gutkovskiy, Feb 02 2024

A375655 Expansion of e.g.f. exp(x^2 + x * exp(x^2/2)).

Original entry on oeis.org

1, 1, 3, 10, 37, 186, 931, 5608, 36345, 252892, 1961011, 15811896, 139137373, 1286591320, 12584565267, 130564271776, 1410581283121, 16095825151248, 190917669584035, 2366869021623712, 30550349329738581, 408806590130340256, 5688859328729212483

Offset: 0

Seiichi Manyama, Aug 22 2024

nonn

Cf. A080108, A375656.

Cf. A375633.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2+x*exp(x^2/2))))

a(n) = n!*sum(k=0, n\2, ((n-2*k+2)/2)^k/(k!*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/2)} ((n-2*k+2)/2)^k / (k! * (n-2*k)!).

A375656 Expansion of e.g.f. exp(x^3 + x * exp(x^3/6)).

Original entry on oeis.org

1, 1, 1, 7, 29, 81, 541, 3781, 18537, 129529, 1171961, 8446131, 66198661, 683784817, 6492131829, 59303102041, 664191218321, 7659196889841, 82391350746097, 991483941558079, 13066764825298221, 164001743446274161, 2139651772557011021, 30946063565684912877

Offset: 0

Seiichi Manyama, Aug 22 2024

nonn

Cf. A080108, A375655.

Cf. A375634.

With[{nn=30},CoefficientList[Series[Exp[x^3+x Exp[x^3/6]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3+x*exp(x^3/6))))

a(n) = n!*sum(k=0, n\3, ((n-3*k+6)/6)^k/(k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} ((n-3*k+6)/6)^k / (k! * (n-3*k)!).

A224786 G.f. satisfies: A(x) = Sum_{n>=0} x^n / (A(x) - n*x)^n.

Original entry on oeis.org

1, 1, 1, 2, 6, 23, 110, 607, 3742, 25324, 185566, 1457998, 12195992, 108010446, 1008224881, 9883048933, 101418491070, 1086613660608, 12126900841444, 140682966122152, 1693340044490513, 21111988598271746, 272228110567491910, 3625334790162237116

Offset: 0

Paul D. Hanna, Apr 17 2013

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 110*x^6 + 607*x^7 +...
where, by definition,
A(x) = 1 + x/(A(x) - x) + x^2/(A(x) - 2*x)^2 + x^3/(A(x) - 3*x)^3 + x^4/(A(x) - 4*x)^4 + x^5/(A(x) - 5*x)^5 +....
Also, the g.f. satisfies:
A(x) = 1 + x/A(x) + 2*x^2/A(x)^2 + 6*x^3/A(x)^3 + 23*x^4/A(x)^4 + 104*x^5/A(x)^5 + 537*x^6/A(x)^6 +...+ A080108(n)*x^n/A(x)^n +...

Cf. A211207, A099947, A080108.

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

G.f. satisfies: A(x) = 1 + G(x/A(x)) where G(x) is the g.f. of A080108, where A080108(n) = Sum_{k=1..n} k^(n-k)*C(n-1,k-1).

cp's OEIS Frontend

A245835 E.g.f.: exp( x*(2 + exp(3*x)) ).

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A355471 Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.

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A367874 Expansion of e.g.f. exp(x * (2 + exp(x))).

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A196794 a(n) = Sum_{k=0..n} binomial(n,k)*2^k*(k+1)^(n-k).

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A116072 Central terms of triangle A116071, which equals Pascal's triangle to the matrix power of Pascal's triangle.

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A196795 a(n) = Sum_{k=0..n} binomial(n,k)*3^k*(k+1)^(n-k).

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A367875 Expansion of e.g.f. exp(x * (3 + exp(x))).

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A375655 Expansion of e.g.f. exp(x^2 + x * exp(x^2/2)).

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A245835 E.g.f.: exp( x(2 + exp(3x)) ).

A196794 a(n) = Sum_{k=0..n} binomial(n,k)2^k(k+1)^(n-k).

A196795 a(n) = Sum_{k=0..n} binomial(n,k)3^k(k+1)^(n-k).