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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":"f400c35c913c23086db9d5efb12347ef57331c2cb486cd33be6c4dba74b9cd8a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f46f66d0d087b029","role":"sequence","name":"Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).","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:      6,       16,        3;\n  1:     20,       90,       54,         4;\n  2:     70,      448,      504,       128,         5;\n  3:    252,     2100,     3600,      1800,       250,         6;\n  4:    924,     9504,    22275,     17600,      4950,       432,        7;\n  5:   3432,    42042,   126126,    140140,     63700,     11466,      686,        8;\n  6:  12870,   183040,   672672,    978432,    637000,    188160,    23520,     1024,       9;\n  7:  48620,   787644,  3436992,   6237504,   5397840,   2313360,   479808,    44064,    1458,     10;\n  8: 184756,  3359200, 17005950,  37209600,  40698000,  23442048,  7122150,  1094400,   76950,   2000,   11;\n  9: 705432, 14226212, 82074300, 211048200, 281397600, 206827236, 85164156, 19311600, 2286900, 127050, 2662, 12;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+14*n^2+12*n+(4*n^4+30*n^3+84*n^2+104*n+48)*x)*P(n,x) = (14*n^3+42*n^2+30*n-2+(15*n^4+130*n^3+349*n^2+374*n+140)*x+(12*n^4+99*n^3+290*n^2+361*n+162)*x^2)*P(n-1,x) + (n^2+3*n+2+(n^3+13*n^2+22*n+6)*x+(7*n^3+43*n^2+102*n+82)*x^2-(8*n^3+57*n^2+127*n+90)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 16*x + 3*x^2.","mathematica":"P[0, x_] := 6 + 16*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*P[n - 1, x] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*D[P[n - 1, x], x])/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 6 + 16*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^3 + 42*n^2 + 30*n - 2 + (15*n^4 + 130*n^3 + 349*n^2 + 374*n + 140)*x + (12*n^4 + 99*n^3 + 290*n^2 + 361*n + 162)*x^2)*v[i - 1] + (n^2 + 3*n + 2 + (n^3 + 13*n^2 + 22*n + 6)*x + (7*n^3 + 43*n^2 + 102*n + 82)*x^2 - (8*n^3 + 57*n^2 + 127*n + 90)*x^3)*deriv(v[i - 1], x))/(4*n^3 + 14*n^2 + 12*n + (4*n^4 + 30*n^3 + 84*n^2 + 104*n + 48)*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-f46f66d0d087b029","steps":["\u03b8+1"],"edges":["e-9e0fa588cb146211"],"text":"A110608 \u2192 \u03b8+1"},"headline":"A110608 \u2192 \u03b8+1","chain":null}