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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-218c16a31928c5ea"]},{"id":191347,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-218c16a31928c5ea"]},{"id":191348,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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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":"0f1250d71838f81f6f60ff742e8ce5ad7215e882c309e9fa3a8f0419914f7265","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-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.","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:         4,           2;\n  2:        40,          44,             4;\n  3:       396,        1164,           412,             8;\n  4:      3692,       28508,         23444,          3740,            16;\n  5:     33596,      599972,       1132816,        417460,         33724,            32;\n  6:    303324,    11379628,      44620336,      35180120,       7054484,        303644,           64;\n  7:   2732284,   202395284,    1497157620,    2328876760,     964933172,     116224500,      2733052,         128;\n  8:  24596188,  3459635708,   44937322780,  126192685444,   99979525940,   24711720292,   1889691412,    24597980,       256;\n  9: 221378748, 57690799716, 1249502802472, 5892682174316, 8228639215608, 3812976052732, 608065672424, 30505736628, 221382844, 512;\n","recurrence_sentence":"The row polynomials satisfy (7*n-19+14*x)*P(n,x) = -(126*n-216+(196*n^2-966*n+1332)*x+(343*n^3-2450*n^2+5866*n-5172)*x^2+(1029*n^2-4900*n+5572)*x^3+(686*n-1176)*x^4)*P(n-2,x) - ((98*n-168)*x+(343*n^2-1617*n+1960)*x^2-(343*n^2-2548*n+4242)*x^3-(1029*n-3136)*x^4-686*x^5)*d/dx P(n-2,x) + (77*n-195+(98*n^2-483*n+743)*x+(245*n-616)*x^2+98*x^3)*P(n-1,x) + ((49*n-133)*x-(49*n-231)*x^2-98*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(126*n - 216 + (196*n^2 - 966*n + 1332)*x + (343*n^3 - 2450*n^2 + 5866*n - 5172)*x^2 + (1029*n^2 - 4900*n + 5572)*x^3 + (686*n - 1176)*x^4)*P[n - 2, x] - ((98*n - 168)*x + (343*n^2 - 1617*n + 1960)*x^2 - (343*n^2 - 2548*n + 4242)*x^3 - (1029*n - 3136)*x^4 - 686*x^5)*D[P[n - 2, x], x] + (77*n - 195 + (98*n^2 - 483*n + 743)*x + (245*n - 616)*x^2 + 98*x^3)*P[n - 1, x] + ((49*n - 133)*x - (49*n - 231)*x^2 - 98*x^3)*D[P[n - 1, x], x])/(7*n - 19 + 14*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(126*n - 216 + (196*n^2 - 966*n + 1332)*x + (343*n^3 - 2450*n^2 + 5866*n - 5172)*x^2 + (1029*n^2 - 4900*n + 5572)*x^3 + (686*n - 1176)*x^4)*v[i - 2] - ((98*n - 168)*x + (343*n^2 - 1617*n + 1960)*x^2 - (343*n^2 - 2548*n + 4242)*x^3 - (1029*n - 3136)*x^4 - 686*x^5)*deriv(v[i - 2], x) + (77*n - 195 + (98*n^2 - 483*n + 743)*x + (245*n - 616)*x^2 + 98*x^3)*v[i - 1] + ((49*n - 133)*x - (49*n - 231)*x^2 - 98*x^3)*deriv(v[i - 1], x))/(7*n - 19 + 14*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-5b7da9f56b3380cb","steps":["D+1"],"edges":["e-218c16a31928c5ea"],"text":"A257617 \u2192 D+1"},"headline":"A257617 \u2192 D+1","chain":null}