Search: a225925 -id:a225925
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A215603
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O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^2) - sigma(n^2)) * (-x)^n/n ).
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+10
7
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1, 2, -2, 2, 10, -10, 6, 10, -22, 58, -58, 10, 114, -210, 270, -242, 74, 382, -930, 1474, -1542, 1010, 446, -2798, 5682, -7718, 8030, -5182, -998, 11126, -23802, 35626, -42246, 39450, -20810, -15546, 69514, -133770, 194918, -234106, 227410, -147706, -19738, 282234
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OFFSET
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0,2
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COMMENTS
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Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n) is the sum of divisors of n.
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LINKS
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FORMULA
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EXAMPLE
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O.g.f.: A(x) = 1 + 2*x - 2*x^2 + 2*x^3 + 10*x^4 - 10*x^5 + 6*x^6 + 10*x^7 +...
where
log(A(x)) = 2*x - 8*x^2/2 + 26*x^3/3 - 32*x^4/4 + 62*x^5/5 - 104*x^6/6 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -A054785(n^2)*(-x)^n/n +...
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, -(sigma(2*m^2)-sigma(m^2))*(-x)^m/m)+x^2*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A225957
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O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^3) - sigma(n^3)) * (-x)^n/n ).
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+10
4
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1, 2, -6, 12, 38, -108, 148, 168, -922, 2294, -2656, -1732, 17908, -44516, 60896, -6936, -206474, 650848, -1181394, 1146324, 865832, -6609592, 16632596, -26643544, 22498916, 23275482, -144152248, 349896736, -563311472, 532552508, 233516176, -2378435472, 6264582710
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n), the sum of the divisors of n.
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LINKS
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FORMULA
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O.g.f.: exp( Sum_{n>=1} -A054785(n^3)*(-x)^n/n ).
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EXAMPLE
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O.g.f.: A(x) = 1 + 2*x - 6*x^2 + 12*x^3 + 38*x^4 - 108*x^5 + 148*x^6 + 168*x^7 +...
where
log(A(x)) = 2*x - 8*x^2/2 + 26*x^3/3 - 32*x^4/4 + 62*x^5/5 - 104*x^6/6 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -(-1)^n*A054785(n^3)*x^n/n +...
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, -(sigma(2*m^3)-sigma(m^3))*(-x)^m/m)+x^2*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A225958
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O.g.f.: exp( Sum_{n>=1} (sigma(2*n^3) - sigma(n^3)) * x^n/n ).
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+10
4
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1, 2, 10, 44, 134, 468, 1524, 4584, 13862, 40566, 114880, 321052, 879092, 2360156, 6248864, 16297384, 41902454, 106437600, 267149022, 662979572, 1628437160, 3960377672, 9541519732, 22786066280, 53958062564, 126750346970, 295476011176, 683776368416, 1571299804688
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Compare to the Jacobi theta_3 function:
1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
Here sigma(n) = A000203(n), the sum of the divisors of n.
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LINKS
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FORMULA
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O.g.f.: exp( Sum_{n>=1} A054785(n^3)*x^n/n ).
Logarithmic derivative equals A225959.
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EXAMPLE
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O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 44*x^3 + 134*x^4 + 468*x^5 + 1524*x^6 +...
where
log(A(x)) = 2*x + 8*x^2/2 + 26*x^3/3 + 32*x^4/4 + 62*x^5/5 + 104*x^6/6 + 114*x^7/7 + 128*x^8/8 + 242*x^9/9 + 248*x^10/10 + 266*x^11/11 +...+ A054785(n^3)*x^n/n +...
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^3)-sigma(m^3))*x^m/m)+x^2*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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