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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":"e1e75b8fbbf299ae0ce8f171fbdb28de5cab3187c1f24704ddf9a70bd8761977","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-216193c52744946f","role":"sequence","name":"basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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,      6;\n  2:    10,     24,      9;\n  3:    28,     96,     72,     12;\n  4:    76,    336,    378,    144,     15;\n  5:   208,   1128,   1692,   1008,    240,     18;\n  6:   568,   3648,   6840,   5568,   2130,    360,    21;\n  7:  1552,  11520,  25920,  26784,  14220,   3888,   504,   24;\n  8:  4240,  35712,  93744, 117504,  80100,  30672,  6426,  672,  27;\n  9: 11584, 109152, 327456, 482688, 402480, 199368, 58716, 9888, 864, 30;\n","recurrence_sentence":"The row polynomials satisfy -(48-(12*n-96)*x-(12*n^2+18*n-64)*x^2-(3*n^3+12*n^2+n-16)*x^3-(n^2-n)*x^4)*P(n,x) = -(144*n+48+(108*n^2+420*n+96)*x+(18*n^3+186*n^2+372*n+64)*x^2+(15*n^3+78*n^2+95*n+16)*x^3+(2*n^2-2*n)*x^4)*P(n-1,x) + (144+(144*n+648)*x+(36*n^2+468*n+1092)*x^2+(72*n^2+528*n+846)*x^3+(39*n^2+231*n+288)*x^4+(3*n^2+28*n+31)*x^5+(n+1)*x^6)*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[(-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*D[P[n - 1, x], x])/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)]];\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] = (-(144*n + 48 + (108*n^2 + 420*n + 96)*x + (18*n^3 + 186*n^2 + 372*n + 64)*x^2 + (15*n^3 + 78*n^2 + 95*n + 16)*x^3 + (2*n^2 - 2*n)*x^4)*v[i - 1] + (144 + (144*n + 648)*x + (36*n^2 + 468*n + 1092)*x^2 + (72*n^2 + 528*n + 846)*x^3 + (39*n^2 + 231*n + 288)*x^4 + (3*n^2 + 28*n + 31)*x^5 + (n + 1)*x^6)*deriv(v[i - 1], x))/(-48 + (12*n - 96)*x + (12*n^2 + 18*n - 64)*x^2 + (3*n^3 + 12*n^2 + n - 16)*x^3 + (n^2 - n)*x^4)); 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-eac80c2bf8e566f5","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-237035d3e801d2dc","e-e38d884eef8c1a24"],"text":"A249250 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A249250 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}