A251685 G.f.: Product_{n>=1} [1 + (n+1)*x^n + (n+2)*x^(n+1) + (n+3)*x^(n+2) + (n+4)*x^(n+3) +...].

1, 2, 6, 18, 45, 108, 252, 578, 1270, 2716, 5678, 11678, 23664, 47182, 92538, 178892, 341481, 644648, 1205062, 2231304, 4092646, 7437680, 13398520, 23939558, 42451586, 74754652, 130777182, 227346498, 392806891, 674630766, 1151926416, 1955909898, 3303296389, 5550556238, 9281646642

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Formula

G.f.: Product_{n>=1} (1 + ((n+1)*x^n - n*x^(n+1))/(1-x)^2).

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 45*x^4 + 108*x^5 + 252*x^6 + 578*x^7 +
which equals the infinite product:
A(x) = (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 +...) * (1 + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 +...) * (1 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 +...) * (1 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7 +...) * (1 + 6*x^5 + 7*x^6 + 8*x^7 + 9*x^8 +...) *...
Illustrate A(x) = Product_{n>=1} (1 + ((n+1)*x^n - n*x^(n+1))/(1-x)^2):
A(x) = (1 + (2*x-x^2)/(1-x)^2) * (1 + (3*x^2-2*x^3)/(1-x)^2) * (1 + (4*x^3-3*x^4)/(1-x)^2) * (1 + (5*x^4-4*x^5)/(1-x)^2) * (1 + (6*x^5-5*x^6)/(1-x)^2) * (1 + (7*x^6-6*x^7)/(1-x)^2) * (1 + (8*x^7-7*x^8)/(1-x)^2) *...
The logarithm of the n-th factor (1 + ((n+1)*x^n - n*x^(n+1))/(1-x)^2) begins:
n=1: 2*x + 2*x^2/2 + 2*x^3/3 + 2*x^4/4 + 2*x^5/5 + 2*x^6/6 +...
n=2: 6*x^2/2 + 12*x^3/3 + 2*x^4/4 - 30*x^5/5 - 42*x^6/6 + 42*x^7/7 +...
n=3: 12*x^3/3 + 20*x^4/4 + 30*x^5/5 - 6*x^6/6 - 84*x^7/7 - 220*x^8/8 +...
n=4: 20*x^4/4 + 30*x^5/5 + 42*x^6/6 + 56*x^7/7 - 28*x^8/8 - 180*x^9/9 +...
n=5: 30*x^5/5 + 42*x^6/6 + 56*x^7/7 + 72*x^8/8 + 90*x^9/9 - 70*x^10/10 +...
n=6: 42*x^6/6 + 56*x^7/7 + 72*x^8/8 + 90*x^9/9 + 110*x^10/10 +...
n=7: 56*x^7/7 + 72*x^8/8 + 90*x^9/9 + 110*x^10/10 + 132*x^11/11 +...
n=8: 72*x^8/8 + 90*x^9/9 + 110*x^10/10 + 132*x^11/11 + 156*x^12/12 +...
n=9: 90*x^9/9 + 110*x^10/10 + 132*x^11/11 + 156*x^12/12 +...
the coefficients of which may form a table to illustrate their behavior:
n=1: [  2,   2,   2,   2,   2,   2,   2,   2,    2,    2,    2, ...];
n=2: [  6,  12,   2, -30, -42,  42, 194, 138, -414, -990,  -46, ...];
n=3: [ 12,  20,  30,  -6, -84,-220,-240,  60,  990, 2222, 2496, ...];
n=4: [ 20,  30,  42,  56, -28,-180,-420,-770, -754,   52, 2240, ...];
n=5: [ 30,  42,  56,  72,  90, -70,-330,-714,-1248,-1960,-1800, ...];
n=6: [ 42,  56,  72,  90, 110, 132,-138,-546,-1120,-1890,-2888, ...];
n=7: [ 56,  72,  90, 110, 132, 156, 182,-238, -840,-1656,-2720, ...];
n=8: [ 72,  90, 110, 132, 156, 182, 210, 240, -376,-1224,-2340, ...];
n=9: [ 90, 110, 132, 156, 182, 210, 240, 272,  306, -558,-1710, ...];
n=10:[110, 132, 156, 182, 210, 240, 272, 306,  342,  380, -790, ...]; ...
From this, can one obtain a formula for the logarithmic series:
log(A(x)) = 2*x + 8*x^2/2 + 26*x^3/3 + 44*x^4/4 + 62*x^5/5 + 80*x^6/6 + 184*x^7/7 + 236*x^8/8 + 170*x^9/9 - 292*x^10/10 - 306*x^11/11 + 1508*x^12/12 +...

Prog

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

Crossrefs

Sequence in context: A319415 A230137 A120414 * A341490 A308305 A054136

Adjacent sequences: A251682 A251683 A251684 * A251686 A251687 A251688

Keyword

nonn

Author

Paul D. Hanna, Feb 14 2015

Status

approved