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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"bc5000885f9c1c95a620c80745c851284182134b45270b1c639429b8dd6eb1ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-f45d29a8d57e6cc8","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","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-f45d29a8d57e6cc8","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","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:                -9,                  1;\n  1:               190,                -29,                1;\n  2:             -5800,               1070,              -59,               1;\n  3:            234000,             -48900,             3440,             -99,             1;\n  4:         -11760000,            2690000,          -221500,            8400,          -149,            1;\n  5:         708000000,         -173660000,         16015000,         -726500,         17350,         -209,          1;\n  6:      -49680000000,        12891600000,      -1296960000,        66955000,      -1942500,        31990,       -279,         1;\n  7:     3981600000000,     -1082772000000,     116810800000,     -6660710000,     222530000,     -4503800,      54320,      -359,      1;\n  8:  -358848000000000,    101561760000000,  -11608876000000,    716951600000,  -26708010000,    628194000,   -9395400,     86640,   -449,    1;\n  9: 35925120000000000, -10525982400000000, 1263630600000000, -83371320000000, 3389997500000, -89572770000, 1568280000, -18063000, 131550, -549, 1;\n","recurrence_sentence":"The row polynomials satisfy (10*n-1-x)*P(n,x) = -(100*n^2+90*n-(20*n+9)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -9 + x.","mathematica":"P[0, x_] := -9 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(100*n^2 + 90*n - (20*n + 9)*x + x^2)*P[n - 1, x])/(10*n - 1 - x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = -9 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(100*n^2 + 90*n - (20*n + 9)*x + x^2)*v[i - 1])/(10*n - 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-4f6e747b794bcc2e","steps":["x \u21a6 (\u22121/5)x+1","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-040c826eba4e7d74","e-cdcdb3aca64a476d"],"text":"A028338 \u2192 x \u21a6 (\u22121/5)x+1 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A028338 \u2192 x \u21a6 (\u22121/5)x+1 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}