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  34,      48,       24;\n  3: 1,   20,   132,     416,      688,       576,        192;\n  4: 1,   38,   414,    2192,     6616,     12000,      12960,       7680,        1920;\n  5: 1,   72,  1200,    9520,    44040,    129312,     249184,     314880,      251520,      115200,        23040;\n  6: 1,  138,  3378,   37792,   246360,   1040112,    2996176,    6026496,     8481984,     8198400,      5187840,      1935360,       322560;\n  7: 1,  268,  9436,  143584,  1256080,   7150528,   28254016,   80253952,   166667008,   253854720,    280691712,    219340800,    114831360,     36126720,      5160960;\n  8: 1,  526, 26422,  534448,  6080200,  44840992,  231018592,  868063744,  2439304576,  5192901120,   8403141120,  10276761600,   9344563200,   6123479040,   2732728320,    743178240,    92897280;\n  9: 1, 1040, 74520, 1974000, 28586280, 265839264, 1727332320, 8240939520, 29766314880, 82936793600, 180085287424, 305735086080, 404516444160, 412765839360, 318459617280, 179601408000, 69827788800, 16721510400, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1-(2*n-9)*x-(8*n-27)*x^2-(12*n-37)*x^3-(8*n-24)*x^4-(2*n-6)*x^5)*P(n-1,x) + (x+6*x^2+14*x^3+16*x^4+9*x^5+2*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 - (2*n - 9)*x - (8*n - 27)*x^2 - (12*n - 37)*x^3 - (8*n - 24)*x^4 - (2*n - 6)*x^5)*P[n - 1, x] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (2*n - 9)*x - (8*n - 27)*x^2 - (12*n - 37)*x^3 - (8*n - 24)*x^4 - (2*n - 6)*x^5)*v[i - 1] + (x + 6*x^2 + 14*x^3 + 16*x^4 + 9*x^5 + 2*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":null,"headline":"mobius_x_over_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","chain":null}