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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"0ee84651b1eac67b1503a26543741005671a08369b8c30b94d0867cc83c814d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-054881b8a1092a7d","role":"sequence","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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:         5,           4;\n  2:        23,          84,           32;\n  3:       151,         936,         1552,          384;\n  4:      1297,       11144,        32240,        33664,          6144;\n  5:     13773,      150076,       612560,      1129536,        851456,        122880;\n  6:    174183,     2289260,     11986960,     31477824,      42164224,      24735744,       2949120;\n  7:   2555439,    39223568,    249686368,    846806016,    1624368896,    1699311616,     813514752,      82575360;\n  8:  42655905,   747577296,   5589301920,  23159091200,   57621014784,   86633902080,   74116284416,   29913513984,    2642411520;\n  9: 798123285, 15707066292, 134655986784, 657749352064, 2006008872192, 3923459306496, 4835879649280, 3492587962368, 1216934313984, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(16*n^3-40*n^2+24*n)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (16*n^3 - 40*n^2 + 24*n)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 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-849de7332df28657","steps":["basis A013611","D+1"],"edges":["e-3d3e7193ca830199","e-923658125b2354e5"],"text":"A088996 \u2192 basis A013611 \u2192 D+1"},"headline":"A088996 \u2192 basis A013611 \u2192 D+1","chain":null}