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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":"54e1d5109fa70184581f61ae34aaacbee0eec77fbb195af015a4e2abd478c657","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-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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:        0;\n  1:        1;\n  2:       16,           2;\n  3:      143,         286,            3;\n  4:     1166,        8580,         3498,             4;\n  5:     9357,      180004,       270006,         37428,             5;\n  6:    74892,     3242766,     11880264,       6485532,        374460,            6;\n  7:   599179,    54033714,    402426129,     536568172,     135084285,      3595074,           7;\n  8:  4793482,   862035104,  11771195394,   31120953976,   19618658990,   2586105312,    33554374,         8;\n  9: 38347913, 13409874760, 314567562876, 1467525044216, 1834406305270, 629135125752, 46934561660, 306783304, 9;\n","recurrence_sentence":"The row polynomials satisfy (7*n-13)*P(n,x) = -(56*n-48+(98*n^2-364*n+240)*x+(343*n^3-1911*n^2+3206*n-1560)*x^2)*P(n-2,x) - ((49*n-42)*x+(343*n^2-980*n+588)*x^2-(343*n^2-931*n+546)*x^3)*d/dx P(n-2,x) + (63*n-110+(98*n^2-364*n+338)*x)*P(n-1,x) + ((49*n-91)*x-(49*n-91)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(56*n - 48 + (98*n^2 - 364*n + 240)*x + (343*n^3 - 1911*n^2 + 3206*n - 1560)*x^2)*P[n - 2, x] - ((49*n - 42)*x + (343*n^2 - 980*n + 588)*x^2 - (343*n^2 - 931*n + 546)*x^3)*D[P[n - 2, x], x] + (63*n - 110 + (98*n^2 - 364*n + 338)*x)*P[n - 1, x] + ((49*n - 91)*x - (49*n - 91)*x^2)*D[P[n - 1, x], x])/(7*n - 13)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(56*n - 48 + (98*n^2 - 364*n + 240)*x + (343*n^3 - 1911*n^2 + 3206*n - 1560)*x^2)*v[i - 2] - ((49*n - 42)*x + (343*n^2 - 980*n + 588)*x^2 - (343*n^2 - 931*n + 546)*x^3)*deriv(v[i - 2], x) + (63*n - 110 + (98*n^2 - 364*n + 338)*x)*v[i - 1] + ((49*n - 91)*x - (49*n - 91)*x^2)*deriv(v[i - 1], x))/(7*n - 13)); 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-0af3ffe51a3d7181","steps":["D"],"edges":["e-cf870e0fe932cd11"],"text":"A142462 \u2192 D"},"headline":"A142462 \u2192 D","chain":null}