{"id":"n-8874186dc87be571","revision":0,"first_index":0,"fingerprint":"f6-fa1c18ce9fab2e9b","name":"theta_plus_one_squared of basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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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:       1;\n  1:       2,        32;\n  2:       6,       160,        576;\n  3:      24,       832,       5184,        8192;\n  4:     120,      4928,      40896,      114688,      102400;\n  5:     720,     33408,     334080,     1269760,     2048000,      1179648;\n  6:    5040,    256896,    2939904,    13639680,    30208000,     31850496,     12845056;\n  7:   40320,   2216448,   28143360,   150929408,   412160000,    602800128,    449576960,    134217728;\n  8:  362880,  21238272,  293186304,  1758625792,  5596057600,  10173284352,  10610016256,   5905580032,  1358954496;\n  9: 3628800, 223994880, 3314151936, 21756018688, 77943398400, 166199427072, 216875925504, 169919643648, 73383542784, 13421772800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+32*n*x-576*x^2)*P(n-2,x) + (32*n*x^2+256*x^3)*d/dx P(n-2,x) + (2*n+1+32*x)*P(n-1,x) - 32*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 32*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 32*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 32*n*x - 576*x^2)*P[n - 2, x] + (32*n*x^2 + 256*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 32*x)*P[n - 1, x] - 32*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 32*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 32*n*x - 576*x^2)*v[i - 2] + (32*n*x^2 + 256*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 32*x)*v[i - 1] - 32*x^2*deriv(v[i - 1], x)); 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":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-40669049634a7e59","e-bc9122fe36994aa6"],"text":"A137320 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A137320 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}