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A376535
a(n) = (1/5^n) * Sum_{k=0..n^2} ( (binomial(n^2, k) * 2^(n^2-k) * 3^k) (mod 5^n) ).
6
1, 3, 4, 9, 14, 19, 29, 30, 36, 52, 57, 69, 83, 97, 112, 129, 148, 167, 187, 209, 223, 240, 268, 288, 314, 327, 354, 397, 427, 458, 495, 512, 546, 588, 619, 642, 668, 717, 782, 800, 834, 886, 908, 983, 1025, 1069, 1112, 1147, 1197, 1247, 1292, 1369, 1386, 1465, 1553, 1576, 1651
OFFSET
1,2
LINKS
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 (2 + 3*x)^(n^2) modulo 5^n begins:
(2+3*x) (mod 5): [2, 3];
(2+3*x)^4 (mod 5^2): [16, 21, 16, 16, 6];
(2+3*x)^9 (mod 5^3): [12, 37, 97, 27, 92, 13, 13, 53, 98, 58];
(2+3*x)^16 (mod 5^4): [536, 364, 345, 540, 445, 102, 593, 110, 420, 560, 463, 322, 45, 165, 120, 24, 471];
(2+3*x)^25 (mod 5^5): [1307, 575, 975, 275, 3050, 1090, 2325, 2100, 2400, 550, 1320, 2700, 1600, 2400, 1300, 1430, 950, 2225, 2525, 2300, 1035, 2825, 2475, 1775, 2175, 68];
(2+3*x)^36 (mod 5^6): [7986, 9369, 7655, 5135, 6905, 13163, 4357, 7920, 11815, 13470, 2991, 9184, 13075, 150, 5950, 590, 185, 12275, 12925, 275, 15585, 11115, 3200, 15150, 6450, 7769, 2226, 6445, 12315, 9945, 4262, 11318, 11930, 5010, 3880, 10154, 14746];
(2+3*x)^49 (mod 5^7): [61937, 60182, 57177, 54597, 62892, 67792, 42587, 52007, 48227, 30072, 24182, 43377, 69322, 55567, 13437, 7967, 20512, 27557, 21402, 15372, 34587, 32582, 62202, 65372, 67167, 61688, 25318, 67073, 42778, 27608, 27608, 48063, 57643, 62298, 66953, 70943, 34873, 59553, 36433, 30438, 70008, 9713, 47418, 35198, 17853, 42038, 73418, 46923, 46878, 58833];
...
where a(n) equals the sum of row n divided by 5^n:
a(1) = (2 + 3)/5 = 1;
a(2) = (16 + 21 + 16 + 16 + 6)/5^2 = 3;
a(3) = (12 + 37 + 97 + 27 + 92 + 13 + 13 + 53 + 98 + 58)/5^3 = 4;
a(4) = (536 + 364 + 345 + 540 + 445 + 102 + 593 + 110 + 420 + 560 + 463 + 322 + 45 + 165 + 120 + 24 + 471)/5^4 = 9;
a(5) = (1307 + 575 + 975 + 275 + 3050 + 1090 + 2325 + 2100 + 2400 + 550 + 1320 + 2700 + 1600 + 2400 + 1300 + 1430 + 950 + 2225 + 2525 + 2300 + 1035 + 2825 + 2475 + 1775 + 2175 + 68)/5^5 = 14;
a(6) = (7986 + 9369 + 7655 + 5135 + 6905 + 13163 + 4357 + 7920 + 11815 + 13470 + 2991 + 9184 + 13075 + 150 + 5950 + 590 + 185 + 12275 + 12925 + 275 + 15585 + 11115 + 3200 + 15150 + 6450 + 7769 + 2226 + 6445 + 12315 + 9945 + 4262 + 11318 + 11930 + 5010 + 3880 + 10154 + 14746)/5^6 = 19;
...
PROG
(PARI) {a(n) = sum(k=0, n^2, ( binomial(n^2, k) * 2^(n^2-k) * 3^k ) % (5^n) )/5^n}
for(n=1, 60, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 06 2024
STATUS
approved