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the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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:         6,          4;\n  2:        34,         92,          32;\n  3:       208,       1112,        1616,          384;\n  4:      1424,      11832,       33296,        33920,         6144;\n  5:     10944,     125552,      541696,      1056320,       838144,        122880;\n  6:     93888,    1386896,     8189056,     24267200,     36588544,      23924736,       2949120;\n  7:    892224,   16211776,   122518208,    493804800,   1124466944,    1390317568,     776060928,      82575360;\n  8:   9320256,  201718080,  1867198656,   9589767424,  29566159104,   54963548160,   57811779584,   28219539456,    2642411520;\n  9: 106266240, 2675175552, 29403530112, 184366645504, 721888670208, 1810694464512, 2853807276032, 2618308362240, 1137454350336, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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-4d60c3dd8627c01d","steps":["basis A013611","D+1"],"edges":["e-60ae4909e1eb1035","e-4bbf911d82e5357d"],"text":"A094638 \u2192 basis A013611 \u2192 D+1"},"headline":"A094638 \u2192 basis A013611 \u2192 D+1","chain":null}