A155810 -id:A155810 - cp's OEIS frontend

A155200 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2) * x^n/n ), a power series in x with integer coefficients.

1, 2, 10, 188, 16774, 6745436, 11466849412, 80444398636280, 2306003967992402758, 268654794629082985019564, 126765597346260977505891041836, 241678070948246232010898235031930952, 1858395916567787793818891330877931472153500, 57560683587056536617649234722821582390470430186648

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, it appears that for m integer, exp( Sum_{n >= 1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.
This is correct: if b(n) = m^(n^2) then by the little Fermat theorem the Gauss congruences hold: b(n*p^k) == b(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k. Then apply Stanley, Ch. 5, Ex. 5.2(a). - Peter Bala, Dec 25 2019
Conjecture: highest exponent of 2 dividing a(n) = A000120(n) = number of 1's in binary expansion of n, so that a(n)/2^A000120(n) is odd for n >= 0. - Paul D. Hanna, Sep 01 2009

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 +...
log(A(x)) = 2*x + 2^4*x^2/2 + 2^9*x^3/3 + 2^16*x^4/4 + 2^25*x^5/5 +...

R. P. Stanley. Enumerative combinatorics, Vol. 2. Volume 62 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1999.

Muniru A Asiru, Table of n, a(n) for n = 0..56
Sawian Jaidee, Patrick Moss, and Tom Ward, Time-changes preserving zeta functions, arXiv:1809.09199 [math.DS], 2018.

Cf. A155201, A155202, A155810 (triangle), variants: A155203, A155207.

Cf. A000120, A156631, A159034.

seq(coeff(series(exp(add(2^(k^2)*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Dec 19 2018

nmax = 14; Exp[Sum[2^(n^2) x^n/n, {n, 1, nmax}]] + O[x]^nmax // CoefficientList[#, x]& (* Jean-François Alcover, Feb 14 2019 *)

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

{a(n)=if(n==0,1,(1/n)*(2*a(n-1) + sum(k=1,n-1,4^k*a(k)*(2*(k+1)*a(n-1-k) - (n-k)*a(n-k)))))} \\ Paul D. Hanna, Mar 11 2009

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,2^(k^2)*a(n-k)))} \\ Paul D. Hanna, Sep 01 2009

Equals column 0 of triangle A155810.
G.f. satisfies: 2*A(x)*A(4x) + 8*x*A(x)*A'(4x) - A'(x)*A(4x) = 0. - Paul D. Hanna, Feb 24 2009
From Paul D. Hanna, Mar 11 2009: (Start)
The differential equation implies recurrence:
n*a(n) = 2*a(n-1) + Sum_{k=1..n-1} 4^k*a(k)*(2*(k+1)*a(n-1-k) - (n-k)*a(n-k)) for n > 0, with a(0) = 1.
G.f. A(x) generates A156631:
A156631(n) = [x^n] A(x)^(2^n) for n >= 0, where the g.f. of A156631 = Sum_{n >= 0} [Sum_{k >= 1} (2^n*2^k*x)^k/k]^n/n!. (End)
a(n) = (1/n)*Sum_{k = 1..n} 2^(k^2)*a(n-k), a(0) = 1. - Vladeta Jovovic, Feb 04 2009
Euler transform of A159034. - Vladeta Jovovic, Apr 02 2009
a(n) = B_n( 0!*2^(1^2), 1!*2^(2^2), 2!*2^(3^2), ..., (n-1)!*2^(n^2) ) / n!, where B_n() is the complete Bell polynomial. - Max Alekseyev, Oct 10 2014
a(n) ~ 2^(n^2) / n. - Vaclav Kotesovec, Oct 09 2019

A155201 G.f.: A(x) = exp( Sum_{n>=1} (2^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 17, 285, 21747, 7894143, 12593691755, 84961748935779, 2379148487805445513, 273416748863491468927893, 128009274688933686165252807225, 242979449433397149030644307317592609, 1863847996727745781866688849374488247858333, 57652096246331953203644653244501049018464175026133

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, it appears that for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 285*x^3 + 21747*x^4 + 7894143*x^5 +...
log(A(x)) = 3*x + 5^2*x^2/2 + 9^3*x^3/3 + 17^4*x^4/4 + 33^5*x^5/5 +...

Cf. A136516, A155200, A155202, A155810 (triangle), variants: A155204, A155208.

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

Equals row sums of triangle A155810.
a(n) = (1/n)*Sum_{k=1..n} (2^k + 1)^k * a(n-k) for n>0, with a(0)=1.
a(n) = B_n( 0!*(2^1+1)^1, 1!*(2^2+1)^2, 2!*(2^3+1)^3, ..., (n-1)!*(2^n+1)^n ) / n!, where B_n() is the n-th complete Bell polynomial. - Max Alekseyev, Oct 10 2014

A155202 G.f.: A(x) = exp( Sum_{n>=1} (2^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 5, 119, 12783, 5739069, 10426379903, 76135573607705, 2234839096465512877, 263966776643953756165279, 125532809982533901346598445525, 240383033223427436734891985275952307

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} (m^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

			G.f.: A(x) = 1 + x + 5*x^2 + 119*x^3 + 12783*x^4 + 5739069*x^5 +...
log(A(x)) = x + 3^2*x^2/2 + 7^3*x^3/3 + 15^4*x^4/4 + 31^5*x^5/5 +...

Cf. A055601, A155200, A155202, A155810 (triangle), variants: A155205, A155209.

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

A156100 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2^nx)^n x^n/n ).

Original entry on oeis.org

1, 1, 3, 7, 25, 113, 741, 7181, 101139, 2089283, 61683087, 2600572391, 156100460443, 13231060891179, 1594932996895155, 270715422001769667, 65209448673400087945, 22130613779988110245993, 10631829612570393072035829, 7207580557759524950136903565, 6902254922678483464065364019049, 9340558535943272871301176019398265, 17831418294195720284498112713266643601

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

Compare to g.f. exp( Sum_{m>=1} (1 + x)^m * x^m/m ) of the Fibonacci sequence.

Conjecture: a(n)^(1/n^2) tends to 2^(1/4). - Vaclav Kotesovec, Oct 17 2020

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 25*x^4 + 113*x^5 + 741*x^6 +...
log(A(x)) = (1 + 2*x)*x + (1 + 2^2*x)^2*x^2/2 + (1 + 2^3*x)^3*x^3/3 +...
log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 65*x^4/4 + 401*x^5/5 + 3521*x^6/6 +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A156101 (log), A155810.

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

G.f.: A(x) = exp(F(x)) where F(x) is the l.g.f. of A156101.

A251657 a(n) = (2^n + 3)^n.

Original entry on oeis.org

1, 5, 49, 1331, 130321, 52521875, 90458382169, 662062621900811, 20248745068443234721, 2548385124666493326171875, 1305282261160894865367626964649, 2701607566979638625212777041914285051, 22497539334127167666989016452232087989410801, 751859086636251929847496735809485838154930419921875

Offset: 0

Paul D. Hanna, Jan 29 2015

nonn

This is a special case of the more general statement:

Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=3.

			E.g.f.: A(x) = 4^0 + 5^1*x + 7^2*x^2/2! + 11^3*x^3/3! + 19^4*x^4/4! + 35^5*x^5/5! + 67^6*x^6/6! + 131^7*x^7/7! +...+ (2^n+3)^n*x^n/n! +...
such that
A(x) = exp(3*x) + 2*exp(3*2*x) + 2^4*exp(3*4*x)*x^2/2! + 2^9*exp(3*8*x)*x^3/3! + 2^16*exp(3*16*x)*x^4/4! +...+ 2^(n^2)*exp(3*2^n*x)*x^n/n! +...

Cf. A055601, A136516, A165327, A155810.

Table[(2^n+3)^n,{n,0,20}] (* Harvey P. Dale, Mar 16 2016 *)

{a(n,q=2,m=1,b=3) =( m*q^n + b)^n}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

{a(n,q=2,m=1,b=3) = sum(k=0,n, binomial(n,k) * b^k * m^(n-k) * (q^n)^(n-k))}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

{a(n,q=2,m=1,b=3) = n!*polcoeff(sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

{a(n,q=2,m=1,b=3) = polcoeff(sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x  +x*O(x^n))^(k+1) ), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

E.g.f.: Sum_{n>=0} 2^(n^2) * exp(3*2^n*x) * x^n/n! = Sum_{n>=0} (2^n + 3)^n * x^n/n!.
O.g.f.: Sum_{n>=0} 2^(n^2) * x^n / (1 + 3*2^n*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n, k) * 3^k * (2^n)^(n-k).
a(n) = Sum_{k=0..n} A155810(k)*3^k.

A155811 a(n) = [x^(n+1)y] exp( Sum_{m>=1} (2^m + y)^m x^m/m ), in which the coefficients of x^n*y^k are integer for n>=k>=0.

Original entry on oeis.org

1, 6, 82, 4452, 1074934, 1082704500, 4411700155252, 72146891831948808, 4724816968764733073446, 1238218148763614236043117508, 1298203457233136135837147642852956

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

Cf. A155810.

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

Column 1 of triangle A155810.

cp's OEIS Frontend

A155200 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2) * x^n/n ), a power series in x with integer coefficients.

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A155201 G.f.: A(x) = exp( Sum_{n>=1} (2^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155202 G.f.: A(x) = exp( Sum_{n>=1} (2^n - 1)^n * x^n/n ), a power series in x with integer coefficients.

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A156100 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2^n*x)^n * x^n/n ).

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A251657 a(n) = (2^n + 3)^n.

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A155811 a(n) = [x^(n+1)*y] exp( Sum_{m>=1} (2^m + y)^m * x^m/m ), in which the coefficients of x^n*y^k are integer for n>=k>=0.

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A156100 G.f.: A(x) = exp( Sum_{n>=1} (1 + 2^nx)^n x^n/n ).

A155811 a(n) = [x^(n+1)y] exp( Sum_{m>=1} (2^m + y)^m x^m/m ), in which the coefficients of x^n*y^k are integer for n>=k>=0.