A376536 - OEIS

A376536

a(n) = (1/3^n) * Sum_{k=0..n^3} ( (binomial(n^3, k) * 2^k) (mod 3^n) ).

1, 4, 5, 33, 63, 112, 179, 260, 369, 502, 659, 868, 1098, 1367, 1703, 2046, 2457, 2895, 3392, 3975, 4591, 5322, 6069, 6899, 7807, 8828, 9836, 10948, 12290, 13440, 14838, 16392, 18046, 19712, 21259, 23381, 25382, 27506, 29679, 32083

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..40.

J. B. Roberts, On Binomial Coefficient Residues, Canadian Journal of Mathematics, Volume 9 1957, pp. 363 - 370.

EXAMPLE

The table of residues of coefficients of x^k in (1 + 2*x)^(n^3) modulo 3^n begins:

(1+2*x) (mod 3): [1, 2];

(1+2*x)^8 (mod 3^2): [1, 7, 4, 7, 4, 1, 1, 7, 4];

(1+2*x)^27 (mod 3^3): [1, 0, 0, 18, 0, 0, 9, 0, 0, 24, 0, 0, 18, 0, 0, 9, 0, 0, 3, 0, 0, 18, 0, 0, 9, 0, 0, 26];

(1+2*x)^64 (mod 3^4): [1, 47, 45, 78, 30, 72, 66, 6, 45, 11, 40, 54, 45, 36, 54, 36, 18, 54, 48, 15, 27, 63, 18, 27, 45, 63, 27, 46, 29, 72, 6, 21, 18, 3, 15, 72, 53, 43, 27, 9, 72, 27, 72, 36, 27, 33, 66, 54, 72, 9, 54, 63, 72, 54, 7, 32, 45, 51, 57, 72, 39, 33, 45, 44, 52];

(1+2*x)^125 (mod 3^5): [1, 7, 139, 220, 55, 232, 46, 106, 211, 140, 89, 182, 56, 95, 62, 140, 224, 128, 78, 60, 150, 42, 132, 168, 213, 195, 42, 65, 212, 44, 233, 119, 122, 47, 140, 242, 238, 127, 115, 214, 94, 46, 184, 100, 196, 171, 225, 198, 36, 9, 144, 9, 63, 36, 177, 24, 60, 222, 177, 159, 204, 132, 6, 12, 84, 210, 183, 228, 3, 174, 84, 48, 72, 18, 45, 207, 234, 99, 234, 180, 207, 95, 179, 83, 83, 203, 8, 158, 26, 38, 79, 229, 127, 145, 16, 13, 160, 13, 181, 165, 183, 93, 201, 111, 75, 30, 48, 201, 148, 64, 160, 214, 175, 208, 31, 82, 232, 17, 200, 95, 131, 53, 119, 71, 65, 176];

...

where a(n) equals the sum of row n divided by 3^n:

a(1) = (1 + 2)/3 = 1;

a(2) = (1 + 7 + 4 + 7 + 4 + 1 + 1 + 7 + 4)/3^2 = 4;

a(3) = (1 + 0 + 0 + 18 + 0 + 0 + 9 + 0 + 0 + 24 + 0 + 0 + 18 + 0 + 0 + 9 + 0 + 0 + 3 + 0 + 0 + 18 + 0 + 0 + 9 + 0 + 0 + 26)/3^3 = 5;

a(4) = (1 + 47 + 45 + 78 + 30 + 72 + 66 + 6 + 45 + 11 + 40 + 54 + 45 + 36 + 54 + 36 + 18 + 54 + 48 + 15 + 27 + 63 + 18 + 27 + 45 + 63 + 27 + 46 + 29 + 72 + 6 + 21 + 18 + 3 + 15 + 72 + 53 + 43 + 27 + 9 + 72 + 27 + 72 + 36 + 27 + 33 + 66 + 54 + 72 + 9 + 54 + 63 + 72 + 54 + 7 + 32 + 45 + 51 + 57 + 72 + 39 + 33 + 45 + 44 + 52)/3^4 = 33;

...

PROG

(PARI) {a(n) = sum(k=0, n^3, (binomial(n^3, k) * 2^k) % (3^n) )/3^n}

for(n=1, 35, print1(a(n), ", "))

(Python)

def A376536(n):

m, r, c, a, b, d = 3**n, n**3, 1, 2, n**3, 1

for k in range(1, r+1):

c += b//d*a%m

a = a*2%m

b *= r-k

d *= k+1