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A106336
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Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.
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5
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1, 1, 1, 2, 5, 11, 25, 64, 169, 442, 1172, 3180, 8730, 24116, 67159, 188568, 532741, 1512695, 4315996, 12369324, 35587923, 102747636, 297601382, 864525312, 2518185362, 7353088206, 21520084301, 63115752910, 185474840912
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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FORMULA
| G.f.: A(x) = (1/x)*serreverse( x*eta(x)/eta(x^2)^2 ).
G.f. satisfies:
(1) A(x) = F(x*A(x)) where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
(2) log(A(x)) = Sum_{n>=1} A106337(n)/n*x^n.
(3) A(x) = Product_{n>=1} (1 + (x*A(x))^n)*(1 - (x*A(x))^(2n)). [From Paul D. Hanna, Oct 23 2010]
(4) A(x) = exp( Sum_{n>=1} (x^n*A(x)^n/(1 + x^n*A(x)^n))/n ). [From Paul D. Hanna, Jun 1 2011]
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EXAMPLE
| G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 64*x^7 +...
A(x) = F(x*A(x)) where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + ...
The radius of convergence equals r = 0.322627632692191133... (A106335)
at which the g.f. converges to A(r) = 1.987369721184684145... (A106334).
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PROG
| (PARI) {a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n+2)/eta(X)^(n+1)/(n+1), n))}
(PARI) {a(n)=if(n<0, 0, polcoeff( sum(k=1, (sqrtint(8*n+1)+1)\2, x^((k^2-k)/2), x*O(x^n))^(n+1)/(n+1), n))}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=prod(m=1, n, (1+(x*A)^m)*(1-(x*A)^(2*m)))); polcoeff(A, n)} [From Paul D. Hanna, Oct 23 2010]
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x*A)^m/(1+(x*A)^m+x*O(x^n))/m))); polcoeff(A, n)} [From Paul D. Hanna, Jun 1 2011]
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CROSSREFS
| Cf. A106333, A106334, A106335, A106337; related: A109085.
Sequence in context: A106805 A094981 A097779 * A047775 A001432 A127075
Adjacent sequences: A106333 A106334 A106335 * A106337 A106338 A106339
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 29 2005
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EXTENSIONS
| Edited by Paul D. Hanna (pauldhanna(AT)juno.com), Jun 01 2011
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