A318636 Expansion of Sum_{n>=1} ( (1 + x^n)^n - 1 ).

1, 2, 3, 5, 5, 9, 7, 14, 10, 20, 11, 31, 13, 35, 25, 45, 17, 74, 19, 70, 56, 77, 23, 161, 26, 104, 111, 154, 29, 261, 31, 222, 198, 170, 56, 536, 37, 209, 325, 496, 41, 623, 43, 605, 626, 299, 47, 1407, 50, 602, 731, 1092, 53, 1305, 517, 1443, 1026, 464, 59, 4002, 61, 527, 1429, 2381, 1352, 2596, 67, 3009, 1840, 2787, 71, 6719, 73, 740, 5378, 4655, 407, 5135, 79, 10118

Offset: 1

Author

Paul D. Hanna, Sep 07 2018

Keywords

• nonn

Examples

			G.f.: A(x) = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 14*x^8 + 10*x^9 + 20*x^10 + 11*x^11 + 31*x^12 + 13*x^13 + 35*x^14 + 25*x^15 + 45*x^16 + ...
such that
A(x) = x + (1 + x^2)^2 - 1 + (1 + x^3)^3 - 1 + (1 + x^4)^4 - 1 + (1 + x^5)^5 - 1 + (1 + x^6)^6 - 1 + (1 + x^7)^7-1 + ...
RELATED SERIES.
The g.f. A(x) equals following series at y = 1:
Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1024 from Paul D. Hanna)

Crossrefs

Cf. A143862, A318637, A318638.

Programs

PARI

{a(n) = polcoeff( sum(m=1,n, (x^m + 1 +x*O(x^n))^m - 1), n)}
for(n=1,100, print1(a(n),", "))

Formula

a(n) = Sum_{d|n} binomial(n/d,d). - Ridouane Oudra, May 02 2019

G.f.: Sum_{k >=1} x^(k^2)/(1-x^k)^(k+1). - Seiichi Manyama, Oct 30 2023