A155201 -id:A155201 - 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

A155204 G.f.: A(x) = exp( Sum_{n>=1} (3^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 58, 7528, 11333974, 173018964568, 25223063625377572, 34295288559321731710864, 429734241619476967064512081894, 49292144502053186639397817183561560472

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 + 4*x + 58*x^2 + 7528*x^3 + 11333974*x^4 + 173018964568*x^5 +...
log(A(x)) = 4*x + 10^2*x^2/2 + 28*x^3/3 + 82^4*x^4/4 + 244^5*x^5/5 +...

Cf. A155203, A155205, A155206, A155812 (triangle), A202989; variants: A155201, A155208.

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

Equals row sums of triangle A155812.

A155810 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (2^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

Original entry on oeis.org

1, 2, 1, 10, 6, 1, 188, 82, 14, 1, 16774, 4452, 490, 30, 1, 6745436, 1074934, 71108, 2602, 62, 1, 11466849412, 1082704500, 43173414, 951300, 13002, 126, 1, 80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1, 2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1, 268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1

Offset: 0

Paul D. Hanna, Feb 04 2009

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,y) = 1 + (2 + y)x + (10 + 6y + y^2)x^2 + (188 + 82y + 14y^2 + y^3)x^3 +...
Triangle begins:
1;
2, 1;
10, 6, 1;
188, 82, 14, 1;
16774, 4452, 490, 30, 1;
6745436, 1074934, 71108, 2602, 62, 1;
11466849412, 1082704500, 43173414, 951300, 13002, 126, 1;
80444398636280, 4411700155252, 104251164804, 1387446246, 11470404, 62538, 254, 1;
2306003967992402758, 72146891831948808, 989785148972932, 7803708940836, 38993810694, 129076164, 292810, 510, 1;
268654794629082985019564, 4724816968764733073446, 36967624172237518088, 169140002768370820, 500466007443108, 1001353593606, 1382564804, 1343434, 1022, 1; ...

Cf. A155200 (column 0), A155201 (row sums), A155811 (column 1).

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n+1,(2^m+y)^m*x^m/m)+x*O(x^n)),n,x),k,y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k.

A155208 G.f.: A(x) = exp( Sum_{n>=1} (4^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 5, 157, 92285, 1091087581, 226287110093405, 788215837483128170845, 45292586018794926904179045725, 42540488665745908362239138191829777245, 649578584556365450465861374646071307864262693725

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 + 5*x + 157*x^2 + 92285*x^3 + 1091087581*x^4 +...
log(A(x)) = 5*x + 17^2*x^2/2 + 65^3*x^3/3 + 257^4*x^4/4 + 1025^5*x^5/5 +...

Cf. A155207, A155209, A155210, variants: A155201, A155204.

{a(n)=polcoeff(exp(sum(m=1,n+1,(4^m+1)^m*x^m/m)+x*O(x^n)),n)}

A156630 G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} (2^n + 2^k)^k*x^k/k ]^n / n!, a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 36, 692, 38186, 10012732, 14013453284, 89892733239928, 2455110210935634790, 278266942487534934333100, 129264916198375365693754194988, 244287539590735476133066282560012360

Offset: 0

Paul D. Hanna, Feb 12 2009

nonn

Compare to these dual g.f.s:
Sum_{n>=0} [ Sum_{k>=1} (2^n+1)^k*x^k/k ]^n/n! (A133991);
Sum_{n>=0} [ Sum_{k>=1} (2^k+1)^k*x^k/k ]^n/n! (A155201);
which, when expanded as power series in x, have only integer coefficients.

			G.f.: A(x) = 1 + 4*x + 36*x^2 + 692*x^3 + 38186*x^4 + 10012732*x^5 +...

Cf. A156631, A133991, A155201.

{a(n)=polcoeff(sum(j=0,n,sum(k=1, n, ((2^j+2^k)*x)^k/k+x*O(x^n))^j/j!),n)}

A156910 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)/(1 - 2^nx)^n x^n/n ).

Original entry on oeis.org

1, 2, 14, 268, 21462, 7872396, 12585797612, 84949155244024, 2379063526056509734, 273414369715003663482380, 128009001272184822673783879332, 242979321424122460096958142064785384

Offset: 0

Paul D. Hanna, Feb 17 2009

nonn

An example of this logarithmic identity at q=2:

Sum_{n>=1} [q^(n^2)/(1 - q^n*x)^n]*x^n/n = Sum_{n>=1} [(1 + q^n)^n - 1]*x^n/n.

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 268*x^3 + 21462*x^4 +...
log(A(x)) = 2/(1-2*x)*x + 2^4/(1-2^2*x)^2*x^2/2 + 2^9/(1-2^3*x)^3*x^3/3 +...
log(A(x)) = (3-1)*x + (5^2-1)*x^2/2 + (9^3-1)*x^3/3 + (17^4-1)*x^4/4 +...

Cf. A155201, A155200.

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

/* As First Differences of A155201: */
{a(n)=polcoeff((1-x)*exp(sum(m=1, n+1, (2^m+1)^m*x^m/m)+x*O(x^n)), n)}

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

Equals the first differences of A155201.

A202516 G.f.: exp( Sum_{n>=1} (2^n + 3^n)^n * x^n/n ).

Original entry on oeis.org

1, 5, 97, 14735, 22208431, 314664801905, 41448076127290195, 50905029765702161210225, 582983891132858366160979787245, 62080074367851800086180277369110042475, 61205889017397342360456211893643596980919936577

Offset: 0

Paul D. Hanna, Dec 20 2011

nonn

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

			G.f.: A(x) = 1 + 5*x + 97*x^2 + 14735*x^3 + 22208431*x^4 +...
where
log(A(x)) = (2+3)*x + (2^2 + 3^2)^2*x^2/2 + (2^3 + 3^3)^3*x^3/3 + (2^4 + 3^4)^4*x^4/4 + (2^5 + 3^5)^5*x^5/5 +...
more explicitly,
log(A(x)) = 5*x + 13^2*x^2/2 + 35^3*x^3/3 + 97^4*x^4/4 + 275^5*x^5/5 +...

Cf. A155200, A155201, A155202, A202517, A326555.

{a(n)=polcoeff(exp(sum(m=1,n,(2^m+3^m)^m*x^m/m)+x*O(x^n)),n)}

A202517 G.f.: exp( Sum_{n>=1} (3^n - 2^n)^n * x^n/n ).

Original entry on oeis.org

1, 1, 13, 2299, 4465027, 83649932869, 14413888012788031, 22412828378864422506133, 312169717565869706933620630009, 38865154523992131836783382601539858727, 43266472789023671032936589458127528396392744933

Offset: 0

Paul D. Hanna, Dec 20 2011

nonn

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

			G.f.: A(x) = 1 + x + 13*x^2 + 2299*x^3 + 4465027*x^4 + 83649932869*x^5 +...
where
log(A(x)) = (3-2)*x + (3^2 - 2^2)^2*x^2/2 + (3^3 - 2^3)^3*x^3/3 + (3^4 - 2^4)^4*x^4/4 + (3^5 - 2^5)^5*x^5/5 +...
more explicitly,
log(A(x)) = x + 5^2*x^2/2 + 19^3*x^3/3 + 65^4*x^4/4 + 211^5*x^5/5 +...

Cf. A202516, A155200, A155201, A155202.

{a(n)=polcoeff(exp(sum(m=1,n,(3^m-2^m)^m*x^m/m)+x*O(x^n)),n)}

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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A155204 G.f.: A(x) = exp( Sum_{n>=1} (3^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

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A155810 Triangle, read by rows, where g.f.: A(x,y) = exp( Sum_{n>=1} (2^n + y)^n * x^n/n ) is a power series in x and y with integer coefficients.

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A155208 G.f.: A(x) = exp( Sum_{n>=1} (4^n + 1)^n * x^n/n ), a power series in x with integer coefficients.

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A156630 G.f.: A(x) = Sum_{n>=0} [ Sum_{k>=1} (2^n + 2^k)^k*x^k/k ]^n / n!, a power series in x with integer coefficients.

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A156910 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x)^n * x^n/n ).

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A202516 G.f.: exp( Sum_{n>=1} (2^n + 3^n)^n * x^n/n ).

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A202517 G.f.: exp( Sum_{n>=1} (3^n - 2^n)^n * x^n/n ).

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A156910 G.f.: A(x) = exp( Sum_{n>=1} 2^(n^2)/(1 - 2^nx)^n x^n/n ).