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 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

%I

%S 1,2,6,18,45,108,252,578,1270,2716,5678,11678,23664,47182,92538,

%T 178892,341481,644648,1205062,2231304,4092646,7437680,13398520,

%U 23939558,42451586,74754652,130777182,227346498,392806891,674630766,1151926416,1955909898,3303296389,5550556238,9281646642

%N 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) +...].

%H Paul D. Hanna, <a href="/A251685/b251685.txt">Table of n, a(n) for n = 0..1000</a>

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

%e 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 +

%e which equals the infinite product:

%e 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 +...) *...

%e Illustrate A(x) = Product_{n>=1} (1 + ((n+1)*x^n - n*x^(n+1))/(1-x)^2):

%e 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) *...

%e The logarithm of the n-th factor (1 + ((n+1)*x^n - n*x^(n+1))/(1-x)^2) begins:

%e n=1: 2*x + 2*x^2/2 + 2*x^3/3 + 2*x^4/4 + 2*x^5/5 + 2*x^6/6 +...

%e 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 +...

%e 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 +...

%e 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 +...

%e 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 +...

%e n=6: 42*x^6/6 + 56*x^7/7 + 72*x^8/8 + 90*x^9/9 + 110*x^10/10 +...

%e n=7: 56*x^7/7 + 72*x^8/8 + 90*x^9/9 + 110*x^10/10 + 132*x^11/11 +...

%e n=8: 72*x^8/8 + 90*x^9/9 + 110*x^10/10 + 132*x^11/11 + 156*x^12/12 +...

%e n=9: 90*x^9/9 + 110*x^10/10 + 132*x^11/11 + 156*x^12/12 +...

%e the coefficients of which may form a table to illustrate their behavior:

%e n=1: [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...];

%e n=2: [ 6, 12, 2, -30, -42, 42, 194, 138, -414, -990, -46, ...];

%e n=3: [ 12, 20, 30, -6, -84,-220,-240, 60, 990, 2222, 2496, ...];

%e n=4: [ 20, 30, 42, 56, -28,-180,-420,-770, -754, 52, 2240, ...];

%e n=5: [ 30, 42, 56, 72, 90, -70,-330,-714,-1248,-1960,-1800, ...];

%e n=6: [ 42, 56, 72, 90, 110, 132,-138,-546,-1120,-1890,-2888, ...];

%e n=7: [ 56, 72, 90, 110, 132, 156, 182,-238, -840,-1656,-2720, ...];

%e n=8: [ 72, 90, 110, 132, 156, 182, 210, 240, -376,-1224,-2340, ...];

%e n=9: [ 90, 110, 132, 156, 182, 210, 240, 272, 306, -558,-1710, ...];

%e n=10:[110, 132, 156, 182, 210, 240, 272, 306, 342, 380, -790, ...]; ...

%e From this, can one obtain a formula for the logarithmic series:

%e 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 +...

%o (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)}

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 14 2015

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Last modified September 26 13:34 EDT 2021. Contains 347668 sequences. (Running on oeis4.)