A059937
Sum of binary numbers with n 1's and two (possibly leading) 0's.
Original entry on oeis.org
0, 7, 45, 186, 630, 1905, 5355, 14308, 36828, 92115, 225225, 540606, 1277874, 2981797, 6881175, 15728520, 35651448, 80215911, 179306325, 398458690, 880803630, 1937768217, 4244635395, 9261022956, 20132658900, 43620761275
Offset: 0
a(2) = 45 since binary sum of 1100 + 1010 + 1001 + 0110 + 0101 + 0011 is 12 + 10 + 9 + 6 + 5 + 3 = 45.
A059938
Sum of binary numbers with n 1's and two (non-leading) 0's.
Original entry on oeis.org
0, 4, 31, 141, 506, 1590, 4593, 12523, 32740, 82908, 204755, 495561, 1179582, 2768818, 6422437, 14745495, 33554312, 75759480, 169869159, 378535765, 838860610, 1849687854, 4060086041, 8875147011, 19327352556, 41943039700
Offset: 0
a(2) = 1100_2 + 1010_2 + 1001_2 = 12 + 10 + 9 = 31.
A356117
T(n, k) = [x^k] (1/2 - x)^(-n) - (1 - x)^(-n).
Original entry on oeis.org
0, 1, 3, 3, 14, 45, 7, 45, 186, 630, 15, 124, 630, 2540, 8925, 31, 315, 1905, 8925, 35770, 128898, 63, 762, 5355, 28616, 128898, 515844, 1891890, 127, 1785, 14308, 85932, 429870, 1891890, 7568484, 28113228, 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725
Offset: 0
Triangle T(n, k) starts:
[0] 0;
[1] 1, 3;
[2] 3, 14, 45;
[3] 7, 45, 186, 630;
[4] 15, 124, 630, 2540, 8925;
[5] 31, 315, 1905, 8925, 35770, 128898;
[6] 63, 762, 5355, 28616, 128898, 515844, 1891890;
[7] 127, 1785, 14308, 85932, 429870, 1891890, 7568484, 28113228;
[8] 255, 4088, 36828, 245640, 1351350, 6487272, 28113228, 112456344, 421717725;
-
ser := series((1/2 - x)^(-n) - (1 - x)^(-n), x, 20):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9);
-
row[n_] := CoefficientList[Series[(1/2 - x)^(-n) - (1 - x)^(-n), {x, 0, n}], x]; row[0] = {0}; Table[row[n], {n, 0, 8}] // Flatten (* Amiram Eldar, Aug 22 2022 *)
A348621
The number of additions required to compute the permanent of general n X n matrices using Ryser's formula without Gray code ordering.
Original entry on oeis.org
0, 4, 21, 82, 275, 836, 2373, 6406, 16647, 41992, 103433, 249866, 593931, 1392652, 3227661, 7405582, 16842767, 38010896, 85196817, 189792274, 420478995, 926941204, 2034237461, 4445962262, 9680453655, 21005074456, 45432700953, 97978941466, 210721832987, 452045307932
Offset: 1
- Herbert John Ryser, Combinatorial Mathematics, volume 14 of Carus Mathematical Monographs. American Mathematical Soc., (1963), pp. 24-28.
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LinearRecurrence[{8,-25,38,-28,8},{0,4,21,82,275},30]
Showing 1-4 of 4 results.