A325289 -id:A325289

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A325294

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

+10
3

1, 1, 2, 5, 17, 73, 368, 2074, 12663, 82236, 561664, 4004815, 29662508, 227413816, 1800063339, 14681764890, 123207630130, 1062547709801, 9407762681632, 85445941932906, 795514580068247, 7587015660017106, 74078917658328970, 740060483734580171, 7560421405484047766, 78939580213645975075, 841942979579094942598, 9168184497787176646141, 101876790751549107815492

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..250

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 73*x^5 + 368*x^6 + 2074*x^7 + 12663*x^8 + 82236*x^9 + 561664*x^10 + 4004815*x^11 + 29662508*x^12 + ...

such that the following series are equal:

B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + ...

B(x) = 1 + x/(1-x) + x^2/(1-x)^4 + x^3/(1-x)^9 + x^4/(1-x)^16 + x^5/(1-x)^25 + x^6/(1-x)^36 + x^7/(1-x)^49 + x^8/(1-x)^64 + ...

where

B(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 + 1730*x^7 + 8889*x^8 + 48829*x^9 + 284858*x^10 + 1755325*x^11 + ... + A178325(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - 1/(1-x +x*O(x^#A))^(m^2)) ), #A) ); A[n+1]}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A178325, A325289, A326423.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 25 2019

STATUS

approved

A326424

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

+10
3

1, 0, 1, 0, 3, 4, 20, 62, 251, 1002, 4295, 19086, 88369, 423957, 2104214, 10783054, 56969183, 309900293, 1733790827, 9965992962, 58801256594, 355808106682, 2206237014216, 14007443494601, 90994768741426, 604395083728629, 4101881493676885, 28426771732773415, 201044377117957190, 1450195412613951590, 10663346917944740350, 79885242459500736025

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

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

EXAMPLE

G.f.: A(x) = 1 + x^2 + 3*x^4 + 4*x^5 + 20*x^6 + 62*x^7 + 251*x^8 + 1002*x^9 + 4295*x^10 + 19086*x^11 + 88369*x^12 + 423957*x^13 + 2104214*x^14 + ...

such that the following series are equal

B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + A(x)^36*x^8 + A(x)^45*x^9 + ...

and

B(x) = 1 + x + (1+x)*x^2 + (1+x)^3*x^3 + (1+x)^6*x^4 + (1+x)^10*x^5 + (1+x)^15*x^6 + (1+x)^21*x^7 + (1+x)^28*x^8 + (1+x)^36*x^9 + ...

where

B(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 27*x^6 + 81*x^7 + 262*x^8 + 910*x^9 + 3363*x^10 + 13150*x^11 + 54135*x^12 + ... + A121690(n-1)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff( sum(m=0, #A, x^m*(1+x +x*O(x^#A))^(m*(m-1)/2) - x^m*Ser(A)^(m*(m+1)/2) ), #A)); A[n+1]}

for(n=0, 35, print1(a(n), ", "))