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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"7cd8528e9c687612035cb44a8a36cbaa243f627c01f4092120f5d278718ee1d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-4f6e747b794bcc2e","role":"sequence","name":"Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:         2,          1;\n  1:         4,          5,          1;\n  2:        18,         27,         10,         1;\n  3:       120,        199,         95,        17,         1;\n  4:      1050,       1865,       1036,       246,        26,        1;\n  5:     11340,      21213,      13089,      3710,       530,       37,       1;\n  6:    145530,     283731,     189470,     60859,     10550,     1009,      50,      1;\n  7:   2162160,    4362447,    3101503,   1095395,    218099,    25613,    1757,     65,    1;\n  8:  36486450,   75795345,   56733336,  21608460,   4782988,   651574,   55384,   2860,   82,   1;\n  9: 689188500, 1468401525, 1147880385, 465218044, 112069124, 17113614, 1700454, 109596, 4416, 101, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+x)*P(n,x) = (4*n^2+2*n-2+(4*n+1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*P[n - 1, x])/(2*n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*v[i - 1])/(2*n + 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-4f6e747b794bcc2e","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-3733e878156287d8"],"text":"A028338 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}