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0)"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          336,           714,            379;\n  1:         4536,         14721,          15897,            5713;\n  2:        54432,        240912,         398475,          292026,           80032;\n  3:       598752,       3406032,        7701813,         8660757,         4846548,         1080325;\n  4:      6158592,      43483392,      126556128,       194702292,       167231238,        76114830,         14354557;\n  5:     60046272,     514942272,     1859594760,      3679470009,      4319414766,      3014218107,       1159428027,       189853438;\n  6:    560431872,    5758376832,    25183907424,     61661018016,     92868160725,     88382031048,      52022530263,     17344782066,      2511177679;\n  7:   5043886848,   61571083392,   320562481824,    945826659648,   1755852557589,   2137120347960,    1711096972029,    871096234389,    256316427399,     33258000301;\n  8:  44019376128,  635224052736,  3886773351552,  13552599939840,  30159831310128,  45068906035998,   46008114589593,  31781839649040,  14250561002442,   3751683617316,   440953665088;\n  9: 374164697088, 6365996582400, 45313782696000, 183914275761216, 480615616038168, 857088503741457, 1070634053880495, 940541706200538, 571073911849818, 228718085727846, 54467017469472, 5849978617585;\n","recurrence_sentence":"The row polynomials satisfy (n+3+(2*n+6)*x+(n+3)*x^2)*P(n,x) = (6*n+48-(18*n-110)*x-(91*n-39)*x^2-(104*n+60)*x^3-(37*n+37)*x^4)*P(n-1,x) + (36*x+145*x^2+219*x^3+147*x^4+37*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 336 + 714*x + 379*x^2.","mathematica":"P[0, x_] := 336 + 714*x + 379*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*P[n - 1, x] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*D[P[n - 1, x], x])/(n + 3 + (2*n + 6)*x + (n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 336 + 714*x + 379*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 48 - (18*n - 110)*x - (91*n - 39)*x^2 - (104*n + 60)*x^3 - (37*n + 37)*x^4)*v[i - 1] + (36*x + 145*x^2 + 219*x^3 + 147*x^4 + 37*x^5)*deriv(v[i - 1], x))/(n + 3 + (2*n + 6)*x + (n + 3)*x^2)); 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":{"root":"n-44b1dbf5d009bf35","steps":["x \u21a6 x/(x+1)"],"edges":["e-846375dacaadf21d"],"text":"A132166 \u2192 x \u21a6 x/(x+1)"},"headline":"A132166 \u2192 x \u21a6 x/(x+1)","chain":null}