A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

1, 47, 2301, 112747, 5524601, 270705447, 13264566901, 649963778147, 31848225129201, 1560563031330847, 76467588535211501, 3746911838225363547, 183598680073042813801, 8996335323579097876247, 440820430855375795936101, 21600201111913414000868947

Offset: 0

Author

Petros Hadjicostas, Sep 24 2019

Keywords

• nonn
• easy

Comments

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1. (Obviously, this is not always true for m = 0).

For m=1 and n=0, p(7^(2*1)*0 + a(1)) = p(47) = 7^(1+1) * 2546.

For m=1 and n=1, p(7^(2*1)*1 + a(1)) = p(96) = 7^(1+1) * 2410496.

For m=1 and n=2, p(7^(2*1)*2 + a(1)) = p(145) = 7^(1+1) * 508344041.

For m=2 and n=0, p(7^(2*2)*0 + a(2)) = p(2301) = 7^(2+1) * 49629361905981812695622866669844910256876089360.

Essentially the same as A052463. - R. J. Mathar, Oct 08 2019

Links

• Colin Barker, Table of n, a(n) for n = 0..500
• G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
• Eric Weisstein's World of Mathematics, Partition Function P Congruences.
• Wikipedia, G. N. Watson.
• Index entries for linear recurrences with constant coefficients, signature (50,-49).

Crossrefs

Cf. A052463, A071746, A213261, A327714, A327582.

Programs

Mathematica

CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 49 x)), {x, 0, 15}], x] (* Michael De Vlieger, Sep 27 2019 *)
LinearRecurrence[{50,-49},{1,47},20] (* Harvey P. Dale, Mar 09 2023 *)

PARI

a(n) = (23 * 7^(2*n) + 1)/24; \\ Michel Marcus, Sep 25 2019

PARI

Vec((1 - 3*x) / ((1 - x)*(1 - 49*x)) + O(x^20)) \\ Colin Barker, Sep 25 2019

Formula

From Colin Barker, Sep 25 2019: (Start)

G.f.: (1 - 3*x) / ((1 - x)*(1 - 49*x)).

a(n) = 50*a(n-1) - 49*a(n-2) for n>1.

(End)