{"id":"n-0d4d6515fe1b63b7","revision":1,"first_index":0,"fingerprint":"f6-b8f53010a9224832","name":"basis_map_A008277 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,        2;\n  2:       6,       14,         4;\n  3:      24,       96,        60,         8;\n  4:     120,      720,       732,       208,        16;\n  5:     720,     6000,      8760,      3960,       640,       32;\n  6:    5040,    55440,    108360,     69000,     17520,     1824,      64;\n  7:   40320,   564480,   1411200,   1182720,    418320,    67872,    4928,    128;\n  8:  362880,  6289920,  19474560,  20563200,   9476880,  2126208,  239232,  12800,   256;\n  9: 3628800, 76204800, 285223680, 367960320, 211710240, 61477920, 9515520, 785664, 32256, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + 2*x)*P[n - 1, x] + 2*x*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; for(n = 1, N, my(i = n + 1); v[i] = (n + 1 + 2*x)*v[i - 1] + 2*x*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 A008277"],"edges":["e-658de135d35c8882"],"text":"A137320 \u2192 basis A008277"},"headline":"A137320 \u2192 basis A008277","chain":null}