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A376534
a(n) = (1/5^n) * Sum_{k=0..n^2} ( (binomial(n^2, k) * 4^k) (mod 5^n) ).
6
1, 2, 3, 8, 13, 21, 25, 30, 41, 46, 59, 74, 79, 96, 102, 126, 147, 166, 180, 195, 228, 247, 271, 289, 301, 333, 357, 395, 417, 451, 476, 511, 552, 586, 604, 662, 668, 721, 755, 776, 830, 884, 921, 947, 1049, 1052, 1085, 1162, 1186, 1230, 1269, 1334, 1405, 1473, 1503, 1575, 1604, 1694
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 (1 + 4*x)^(n^2) modulo 5^n begins:
(1+4*x) (mod 5): [1, 4];
(1+4*x)^4 (mod 5^2): [1, 16, 21, 6, 6];
(1+4*x)^9 (mod 5^3): [1, 36, 76, 1, 6, 24, 64, 74, 74, 19];
(1+4*x)^16 (mod 5^4): [1, 64, 45, 215, 295, 332, 143, 460, 195, 485, 358, 497, 620, 90, 345, 434, 421];
(1+4*x)^25 (mod 5^5): [1, 100, 1675, 325, 900, 1995, 1600, 1300, 2325, 900, 1385, 2725, 2300, 2950, 2525, 2615, 1850, 425, 1450, 1150, 1380, 3100, 550, 2325, 775, 1999];
(1+4*x)^36 (mod 5^6): [1, 144, 10080, 3835, 1555, 8558, 15407, 14120, 1615, 7945, 13931, 15234, 1950, 14400, 7575, 13190, 14560, 8775, 5800, 15400, 8610, 10615, 13325, 10025, 8700, 2329, 9951, 11270, 14490, 13295, 14492, 1643, 2980, 12335, 8030, 1389, 10571];
(1+4*x)^49 (mod 5^7): [1, 196, 18816, 7261, 21506, 71091, 54086, 12006, 17751, 63046, 71236, 72756, 62201, 59096, 5166, 48216, 19211, 57256, 51251, 1421, 8526, 47096, 19591, 20661, 11406, 45624, 72304, 61184, 58364, 39744, 27859, 60739, 78069, 18824, 40029, 35139, 28619, 8549, 56029, 61209, 61209, 51839, 2294, 50549, 48879, 21724, 61904, 70659, 50839, 20094];
...
where a(n) equals the sum of row n divided by 5^n:
a(1) = (1 + 4)/5 = 1;
a(2) = (1 + 16 + 21 + 6 + 6)/5^2 = 2;
a(3) = (1 + 36 + 76 + 1 + 6 + 24 + 64 + 74 + 74 + 19)/5^3 = 3;
a(4) = (1 + 64 + 45 + 215 + 295 + 332 + 143 + 460 + 195 + 485 + 358 + 497 + 620 + 90 + 345 + 434 + 421)/5^4 = 8;
a(5) = (1 + 100 + 1675 + 325 + 900 + 1995 + 1600 + 1300 + 2325 + 900 + 1385 + 2725 + 2300 + 2950 + 2525 + 2615 + 1850 + 425 + 1450 + 1150 + 1380 + 3100 + 550 + 2325 + 775 + 1999)/5^5 = 13;
a(6) = (1 + 144 + 10080 + 3835 + 1555 + 8558 + 15407 + 14120 + 1615 + 7945 + 13931 + 15234 + 1950 + 14400 + 7575 + 13190 + 14560 + 8775 + 5800 + 15400 + 8610 + 10615 + 13325 + 10025 + 8700 + 2329 + 9951 + 11270 + 14490 + 13295 + 14492 + 1643 + 2980 + 12335 + 8030 + 1389 + 10571)/5^6 = 21;
...
PROG
(PARI) {a(n) = sum(k=0, n^2, ( binomial(n^2, k) * 4^k ) % (5^n) )/5^n}
for(n=1, 60, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 06 2024
STATUS
approved