{"id":"n-a9aff647c13fb176","revision":0,"first_index":1,"fingerprint":"f6-90434d8020f51a22","name":"theta_plus_one of basis_map_A013610 of Coefficients of a special case of Poisson-Charlier 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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-14b085b8762b7422"]},{"id":187661,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-14b085b8762b7422"]},{"id":187662,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve 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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"116bd0ef072364196a19d4b5506f05da754cee3628de94fc7bd275f461bcc632","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6c2910cdb87c65c2","role":"sequence","name":"basis_map_A013610 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,        6;\n   3:      5,       30,        27;\n   4:     16,      150,       297,        108;\n   5:     65,      864,      2889,       3024,         405;\n   6:    326,     5766,     28458,      55188,       38070,        1458;\n   7:   1957,    43878,    292788,     860868,     1176120,      613818,        5103;\n   8:  13700,   375150,   3189537,   12765168,    26733645,    28425168,    12140037,       17496;\n   9: 109601,  3562416,  37027530,  189000864,   539386695,   880830288,   771318450,   281335680,      59049;\n  10: 986410, 37226214, 459066852, 2862726408, 10410122430, 23138534358, 31075657452, 23219116560, 7421396418, 196830;\n","recurrence_sentence":"The row polynomials satisfy (6*n-12)*P(n,x) = -(6*n^3-54*n^2+174*n-198+(54*n^3-540*n^2+1782*n-1944)*x+(162*n^3-1782*n^2+5994*n-6318)*x^2+(162*n^3-1944*n^2+6642*n-6804)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(12*n^3-102*n^2+300*n-306)*x+(54*n^3-486*n^2+1458*n-1458)*x^2+(108*n^3-1026*n^2+3132*n-3078)*x^3+(81*n^3-810*n^2+2511*n-2430)*x^4)*d/dx P(n-3,x) - (18*n^2-78*n+90+(108*n^2-468*n+504)*x+(162*n^2-648*n+648)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(9*n^3-63*n^2+144*n-102)*x+(27*n^3-216*n^2+531*n-414)*x^2+(27*n^3-243*n^2+648*n-540)*x^3)*d/dx P(n-2,x) + (12*n^2-36*n+30+(36*n^2-144*n+144)*x)*P(n-1,x) - (n^2-5*n+8+(6*n^2-36*n+54)*x+(9*n^2-63*n+90)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 6*x, P(3,x) = 5 + 30*x + 27*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 6*x;\nP[3, x_] := 5 + 30*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*D[P[n - 3, x], x] - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*D[P[n - 2, x], x] + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*D[P[n - 1, x], x])/(6*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; v[3] = 5 + 30*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-(6*n^3 - 54*n^2 + 174*n - 198 + (54*n^3 - 540*n^2 + 1782*n - 1944)*x + (162*n^3 - 1782*n^2 + 5994*n - 6318)*x^2 + (162*n^3 - 1944*n^2 + 6642*n - 6804)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (12*n^3 - 102*n^2 + 300*n - 306)*x + (54*n^3 - 486*n^2 + 1458*n - 1458)*x^2 + (108*n^3 - 1026*n^2 + 3132*n - 3078)*x^3 + (81*n^3 - 810*n^2 + 2511*n - 2430)*x^4)*deriv(v[i - 3], x) - (18*n^2 - 78*n + 90 + (108*n^2 - 468*n + 504)*x + (162*n^2 - 648*n + 648)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (9*n^3 - 63*n^2 + 144*n - 102)*x + (27*n^3 - 216*n^2 + 531*n - 414)*x^2 + (27*n^3 - 243*n^2 + 648*n - 540)*x^3)*deriv(v[i - 2], x) + (12*n^2 - 36*n + 30 + (36*n^2 - 144*n + 144)*x)*v[i - 1] - (n^2 - 5*n + 8 + (6*n^2 - 36*n + 54)*x + (9*n^2 - 63*n + 90)*x^2)*deriv(v[i - 1], x))/(6*n - 12)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013610","\u03b8+1"],"edges":["e-3699a0e7eb346ca6","e-14b085b8762b7422"],"text":"A046716 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A046716 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}