{"id":"n-aab6e0876ef53df1","revision":0,"first_index":0,"fingerprint":"f6-16caee28acd0a1b9","name":"one_plus_derivative of basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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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":"eb9eae651312fa0a65504a4391d06f331c48e17cf44a55ad23c48fd6853ff293","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-07810086d0e6802d","role":"sequence","name":"basis_map_A013611 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:       10,         8;\n  2:       46,       168,         64;\n  3:      232,      1360,       2112,         512;\n  4:     1352,     10320,      26048,       23552,        4096;\n  5:     9072,     82592,     275200,      407040,      245760,       32768;\n  6:    69264,    717536,    2884096,     5685760,     5632000,     2457600,      262144;\n  7:   594432,   6808192,   31426304,    75378688,   100208640,    71794688,    23855104,    2097152;\n  8:  5672448,  70462080,  362318592,  1005283328,  1636798464,  1581776896,   862453760,  226492416,   16777216;\n  9: 59627520, 792476928, 4447492608, 13830694912, 26200989696, 31172886528, 23011000320, 9902751744, 2113929216, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+16*n*x+64*x^2)*P(n-2,x) + (2*n+1+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 10 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 16*n*x + 64*x^2)*P[n - 2, x] + (2*n + 1 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 16*n*x + 64*x^2)*v[i - 2] + (2*n + 1 + 16*x)*v[i - 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-60c7e36cbe887b7e","steps":["basis A013611","D+1"],"edges":["e-40669049634a7e59","e-fcf730757ad93292"],"text":"A137320 \u2192 basis A013611 \u2192 D+1"},"headline":"A137320 \u2192 basis A013611 \u2192 D+1","chain":null}