A147544 Coefficient of x^n in product (1+x)*Product_{j>=1} (1 + prime(j)*x^j).

3, 5, 14, 28, 57, 126, 239, 436, 830, 1616, 2806, 4975, 8578, 14914, 26825, 45148, 73742, 124446, 205326, 333878, 560588, 903857, 1431876, 2275324, 3633808, 5713071, 9019922, 14309452, 22104630, 34018220, 52450816, 80023048, 122423244, 186079122, 282089161, 431595854, 647808336, 966099832, 1442708500

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

G.f.: -1 + (1+x)*Product_{j>=1} (1 + prime(j)*x^j).

Example

(1+x)*(1+2*x)*(1+3*x^2)*(1+5*x^3)*(1+7*x^4)*(1+11*x^5) ... = 1 + 3*x + 5*x^2 + 14*x^3 + 28*x^4 + 57*x^5 + 126*x^6 + 239*x^7 + 436*x^8 + 830*x^9 + 1616*x^10 + ..., so the sequence begins 3, 5, 14, ...

Mathematica

With[{m=51}, Rest@CoefficientList[Series[(1+x)*Product[(1+Prime[j]*x^j), {j, m+2}], {x, 0, m}], x]] (* G. C. Greubel, Oct 28 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 51); Coefficients(R!( (1+x)*(&*[1+NthPrime(j)*x^j: j in [1..52]])-1 )); // G. C. Greubel, Oct 28 2022
(SageMath)
def A147544_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( (1+x)*product(1+nth_prime(j)*x^j for j in range(1, 53)) ).list()
a=A147544_list(51); a[1:]  # G. C. Greubel, Oct 28 2022

Crossrefs

Cf. A000040, A147541.

Sequence in context: A026777 A007136 A145974 * A192478 A198785 A222380

Adjacent sequences: A147541 A147542 A147543 * A147545 A147546 A147547

Keyword

nonn

Author

Neil Fernandez, Nov 06 2008

Extensions

Terms a(11) onward corrected by G. C. Greubel, Oct 28 2022

Status

approved