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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":"b5448660a5286c19a666936023d6be1fe3ec6aec5016a1859c0b4fe84bdff239","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-ef179a3c7df58b5a","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(4; m,x).","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:   5,     4;\n  2:  15,    48,      9;\n  3:  35,   252,    189,      16;\n  4:  70,   896,   1512,     512,      25;\n  5: 126,  2520,   7560,    5760,    1125,      36;\n  6: 210,  6048,  28350,   38400,   16875,    2160,      49;\n  7: 330, 12936,  87318,  184800,  144375,   41580,    3773,     64;\n  8: 495, 25344, 232848,  709632,  866250,  443520,   90552,   6144,   81;\n  9: 715, 46332, 555984, 2306304, 4054050, 3243240, 1177176, 179712, 9477, 100;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+36*n+(8*n^4+26*n^3-24*n^2)*x+(n^6+4*n^5)*x^2)*P(n,x) = (9*n^2+72*n+144+(8*n^4+58*n^3+188*n^2+120*n-144)*x+(n^6+8*n^5+55*n^4+94*n^3-25*n^2-92*n+24)*x^2+(3*n^6+14*n^5+19*n^4-4*n^3-12*n^2)*x^3)*P(n-1,x) + ((18*n+36)*x+(16*n^3+44*n^2+16*n-48)*x^2+(2*n^5+10*n^4+4*n^3-32*n^2-34*n+12)*x^3-(2*n^5+10*n^4+20*n^3+12*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 + 72*n + 144 + (8*n^4 + 58*n^3 + 188*n^2 + 120*n - 144)*x + (n^6 + 8*n^5 + 55*n^4 + 94*n^3 - 25*n^2 - 92*n + 24)*x^2 + (3*n^6 + 14*n^5 + 19*n^4 - 4*n^3 - 12*n^2)*x^3)*P[n - 1, x] + ((18*n + 36)*x + (16*n^3 + 44*n^2 + 16*n - 48)*x^2 + (2*n^5 + 10*n^4 + 4*n^3 - 32*n^2 - 34*n + 12)*x^3 - (2*n^5 + 10*n^4 + 20*n^3 + 12*n^2)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 36*n + (8*n^4 + 26*n^3 - 24*n^2)*x + (n^6 + 4*n^5)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 + 72*n + 144 + (8*n^4 + 58*n^3 + 188*n^2 + 120*n - 144)*x + (n^6 + 8*n^5 + 55*n^4 + 94*n^3 - 25*n^2 - 92*n + 24)*x^2 + (3*n^6 + 14*n^5 + 19*n^4 - 4*n^3 - 12*n^2)*x^3)*v[i - 1] + ((18*n + 36)*x + (16*n^3 + 44*n^2 + 16*n - 48)*x^2 + (2*n^5 + 10*n^4 + 4*n^3 - 32*n^2 - 34*n + 12)*x^3 - (2*n^5 + 10*n^4 + 20*n^3 + 12*n^2)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 36*n + (8*n^4 + 26*n^3 - 24*n^2)*x + (n^6 + 4*n^5)*x^2)); 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-9516b04042fe64c0","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-6d0521392d648070","e-893ffa62ec292f8d"],"text":"A062264 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A062264 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}