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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"4bca54e244d9f93548bf2820dd161186c9d7cb43c9c5a1276607d78baab8ec28","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:        3,         1;\n  1:        8,         8,          2;\n  2:       30,        53,         31,          6;\n  3:      144,       372,        358,        152,         24;\n  4:      840,      2882,       3933,       2667,        898,        120;\n  5:     5760,     24768,      44168,      41792,      22118,       6204,        720;\n  6:    45360,    235188,     520560,     637387,     466093,     203464,      49068,      5040;\n  7:   403200,   2452320,    6503768,    9821128,    9233418,    5532592,    2062520,    437184,     40320;\n  8:  3991680,  27898704,   86415588,  155667080,  179681367,  137784453,   70172800,  22881132,   4332816,   362880;\n  9: 43545600, 344200320, 1221295392, 2561246976, 3515154256, 3298318872, 2142395946, 950974136, 275999832, 47279520, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n+2+n*x)*P(n,x) = (n^2+4*n+3+(2*n^2+5*n+1)*x+(n^2+n)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + x.","mathematica":"P[0, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 4*n + 3 + (2*n^2 + 5*n + 1)*x + (n^2 + n)*x^2)*P[n - 1, x])/(n + 2 + n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 4*n + 3 + (2*n^2 + 5*n + 1)*x + (n^2 + n)*x^2)*v[i - 1])/(n + 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-7fd16a32c438d522","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-bcc91de7315c2529"],"text":"A059364 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A059364 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}