A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

5, 243, 11905, 583343, 28583805, 1400606443, 68629715705, 3362856069543, 164779947407605, 8074217422972643, 395636653725659505, 19386196032557315743, 949923605595308471405, 46546256674170115098843, 2280766577034335639843305, 111757562274682446352321943

Offset: 0

Author

Petros Hadjicostas, Sep 23 2019

Keywords

• nonn
• easy

Comments

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m+1)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1.

It is well-known that this result is true even for m = 0 (cf. A071746 and the references there).

Examples

			For m=1 and n=0, p(7^(2*1+1)*0 + a(1)) = p(243) = 133978259344888 = 7^2 * 2734250190712.
For m=1 and n=1, p(7^(2*1+1)*1 + a(1)) = p(586) = 224282898599046831034631 = 7^2 * 4577202012225445531319.
		

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. A000041, A071746, A110375, A213261, A327714, A327770.

Programs

PARI

a(n) = (17 * 7^(2*n+1) + 1)/24; \\ Michel Marcus, Sep 25 2019

PARI

Vec((5 - 7*x) / ((1 - x)*(1 - 49*x)) + O(x^15)) \\ Colin Barker, Sep 27 2019

Formula

From Colin Barker, Sep 27 2019: (Start)

G.f.: (5 - 7*x) / ((1 - x)*(1 - 49*x)).

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

(End)