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Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! 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v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*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":"A181543","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   8,     1;\n  3: 1,  27,    27,      1;\n  4: 1,  64,   216,     64,       1;\n  5: 1, 125,  1000,   1000,     125,       1;\n  6: 1, 216,  3375,   8000,    3375,     216,      1;\n  7: 1, 343,  9261,  42875,   42875,    9261,    343,     1;\n  8: 1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;\n  9: 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = -(n^2-4*n+2-(8*n^2-11*n+4)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) + (2*n^2-4*n+2+(4*n^2-8*n+4)*x+(2*n^2-4*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 2, x])/(n^2)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 2])/(n^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A181543","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   8,     1;\n  3: 1,  27,    27,      1;\n  4: 1,  64,   216,     64,       1;\n  5: 1, 125,  1000,   1000,     125,       1;\n  6: 1, 216,  3375,   8000,    3375,     216,      1;\n  7: 1, 343,  9261,  42875,   42875,    9261,    343,     1;\n  8: 1, 512, 21952, 175616,  343000,  175616,  21952,   512,   1;\n  9: 1, 729, 46656, 592704, 2000376, 2000376, 592704, 46656, 729, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where n^2*P(n,x) = -(n^2-4*n+2-(8*n^2-11*n+4)*x)*P(n-1,x) + ((9*n-6)*x-(9*n-6)*x^2)*d/dx P(n-1,x) + (2*n^2-4*n+2+(4*n^2-8*n+4)*x+(2*n^2-4*n+2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*P[n - 1, x] + ((9*n - 6)*x - (9*n - 6)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*P[n - 2, x])/(n^2)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 2 - (8*n^2 - 11*n + 4)*x)*v[i - 1] + ((9*n - 6)*x - (9*n - 6)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 4*n + 2 + (4*n^2 - 8*n + 4)*x + (2*n^2 - 4*n + 2)*x^2)*v[i - 2])/(n^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-b13531ad79e7c629","steps":["\u2299 C(n,k)"],"edges":["e-835093125a0f64b6"],"text":"A008459 \u2192 \u2299 C(n,k)"},"headline":"A181543: Triangle of cubed binomial coefficients, T(n,k) = C(n,k)^3, read by rows.","chain":"A008459 \u2192 \u2299 C(n,k)"}