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0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        5,         3;\n   3:       20,        33,          9;\n   4:       91,       273,        180,         27;\n   5:      497,      2358,       3231,       1080,          81;\n   6:     3209,     21855,      50877,      44253,        8829,         243;\n   7:    23896,    217131,     743247,    1156113,      746739,      106677,         729;\n   8:   201275,   2313933,   10637055,   24578208,    29034369,    15143274,     1749600,       2187;\n   9:  1890809,  26466228,  154093158,  478759572,   841902579,   807935148,   356357070,   35219448,      6561;\n  10: 19599517, 324657675, 2300067090, 9043779546, 21364136451, 30492700209, 24756106626, 9499186386, 824776749, 19683;\n","recurrence_sentence":"The row polynomials satisfy (6*n-12)*P(n,x) = -(7*n^3-62*n^2+197*n-222+(66*n^3-642*n^2+2082*n-2250)*x+(216*n^3-2268*n^2+7452*n-7776)*x^2+(270*n^3-2970*n^2+9774*n-9882)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(12*n^3-102*n^2+300*n-306)*x+(54*n^3-486*n^2+1458*n-1458)*x^2+(108*n^3-1026*n^2+3132*n-3078)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*d/dx P(n-3,x) + (n^3-24*n^2+91*n-98+(9*n^3-171*n^2+612*n-606)*x+(27*n^3-378*n^2+1179*n-1062)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(9*n^3-63*n^2+144*n-102)*x+(27*n^3-216*n^2+531*n-414)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*d/dx P(n-2,x) + (11*n^2-31*n+22+(30*n^2-108*n+90)*x-(9*n^2-63*n+90)*x^2)*P(n-1,x) - (n^2-5*n+8+(6*n^2-36*n+54)*x+(9*n^2-63*n+90)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 5 + 3*x, P(3,x) = 20 + 33*x + 9*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 3*x;\nP[3, x_] := 20 + 33*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(7*n^3 - 62*n^2 + 197*n - 222 + (66*n^3 - 642*n^2 + 2082*n - 2250)*x + (216*n^3 - 2268*n^2 + 7452*n - 7776)*x^2 + (270*n^3 - 2970*n^2 + 9774*n - 9882)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*D[P[n - 3, x], x] + (n^3 - 24*n^2 + 91*n - 98 + (9*n^3 - 171*n^2 + 612*n - 606)*x + (27*n^3 - 378*n^2 + 1179*n - 1062)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*D[P[n - 2, x], x] + (11*n^2 - 31*n + 22 + (30*n^2 - 108*n + 90)*x - (9*n^2 - 63*n + 90)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; v[3] = 20 + 33*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = (-(7*n^3 - 62*n^2 + 197*n - 222 + (66*n^3 - 642*n^2 + 2082*n - 2250)*x + (216*n^3 - 2268*n^2 + 7452*n - 7776)*x^2 + (270*n^3 - 2970*n^2 + 9774*n - 9882)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*deriv(v[i - 3], x) + (n^3 - 24*n^2 + 91*n - 98 + (9*n^3 - 171*n^2 + 612*n - 606)*x + (27*n^3 - 378*n^2 + 1179*n - 1062)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*deriv(v[i - 2], x) + (11*n^2 - 31*n + 22 + (30*n^2 - 108*n + 90)*x - (9*n^2 - 63*n + 90)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*deriv(v[i - 1], x))/(6*n - 12)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013610","D+1"],"edges":["e-3699a0e7eb346ca6","e-cd4b84e658cbfa1e"],"text":"A046716 \u2192 basis A013610 \u2192 D+1"},"headline":"A046716 \u2192 basis A013610 \u2192 D+1","chain":null}