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Triangle read by rows, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 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:       0;\n  1:       2;\n  2:       2;\n  3:      12,       8;\n  4:      60,      48;\n  5:     328,     344,     32;\n  6:    2040,    2760,    480;\n  7:   14432,   24192,   5984,   128;\n  8:  115136,  230496,  73472,  3584;\n  9: 1025632, 2381728, 928032, 71936, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(8*n^2-64*n+128)*x+16*x^2)*P(n-4,x) + (4*n^3-42*n^2+148*n-175-(16*n-56)*x)*P(n-3,x) - (6*n^2-36*n+55-8*x)*P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2, P(3,x) = 12 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 2;\nP[3, x_] := 12 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*P[n - 3, x] - (6*n^2 - 36*n + 55 - 8*x)*P[n - 2, x] + (4*n - 10)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 2; v[4] = 12 + 8*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (8*n^2 - 64*n + 128)*x + 16*x^2)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 - (16*n - 56)*x)*v[i - 3] - (6*n^2 - 36*n + 55 - 8*x)*v[i - 2] + (4*n - 10)*v[i - 1]); 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-60c7e36cbe887b7e","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-28dcf565f3f6cf28","e-13c79e0962832583"],"text":"A137320 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A137320 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}