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L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. 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v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (323*n - 646)*x*v[i - 2])/(n - 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}},{"oeis_id":"A153653","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  2;\n   2: 19,    19;\n   3:  2,   718,        2;\n   4:  2,  6857,     6857,          2;\n   5:  2,  7505,   245628,       7505,            2;\n   6:  2,  8153,  2467944,    2467944,         8153,            2;\n   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;\n   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;\n   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;\n  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.","mathematica":"P[1, x_] := 2;\nP[2, x_] := 19 + 19*x;\nP[3, x_] := 2 + 718*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A153653","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  2;\n   2: 19,    19;\n   3:  2,   718,        2;\n   4:  2,  6857,     6857,          2;\n   5:  2,  7505,   245628,       7505,            2;\n   6:  2,  8153,  2467944,    2467944,         8153,            2;\n   7:  2,  8801,  4900212,   84273732,      4900212,         8801,          2;\n   8:  2,  9449,  7542432,  886319856,    886319856,      7542432,       9449,        2;\n   9:  2, 10097, 10394604, 2476630764,  28993055148,   2476630764,   10394604,    10097,     2;\n  10:  2, 10745, 13456728, 4923230904, 317750999400, 317750999400, 4923230904, 13456728, 10745, 2;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (323*n-969)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 19 + 19*x, P(3,x) = 2 + 718*x + 2*x^2.","mathematica":"P[1, x_] := 2;\nP[2, x_] := 19 + 19*x;\nP[3, x_] := 2 + 718*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (323*n - 969)*x*P[n - 2, x])/(n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 19 + 19*x; v[3] = 2 + 718*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (323*n - 969)*x*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-71e25d00553f2c01","steps":[],"edges":[],"text":"A153653"},"headline":"A153653: Triangle T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.","chain":null}