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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":"d74352a1c64edf12e1a3817c471f0c0bb45170047d71969444bf13cbd0f1d26d","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-74b50e0ea94d74b6","role":"sequence","name":"basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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:    3,     4;\n  2:    8,    20,      9;\n  3:   21,    76,     66,     16;\n  4:   55,   260,    333,    160,     25;\n  5:  144,   840,   1422,   1024,    325,     36;\n  6:  377,  2616,   5508,   5296,   2555,    588,    49;\n  7:  987,  7940,  19998,  24080,  15650,   5544,   980,    64;\n  8: 2584, 23644,  69327, 100352,  82175,  39528, 10878,  1536,   81;\n  9: 6765, 69380, 232134, 392640, 388225, 236580, 89040, 19776, 2295, 100;\n","recurrence_sentence":"The row polynomials satisfy (20*n+(20*n^2+14*n)*x+(8*n^3+4*n^2)*x^2)*P(n,x) = (80*n-20+(100*n^2+126*n-44)*x+(32*n^3+128*n^2+40*n-28)*x^2+(24*n^3+36*n^2-8*n-4)*x^3)*P(n-1,x) - (25+(50*n+80)*x+(20*n^2+130*n+93)*x^2+(36*n^2+112*n+46)*x^3+(16*n^2+32*n+8)*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[((80*n - 20 + (100*n^2 + 126*n - 44)*x + (32*n^3 + 128*n^2 + 40*n - 28)*x^2 + (24*n^3 + 36*n^2 - 8*n - 4)*x^3)*P[n - 1, x] - (25 + (50*n + 80)*x + (20*n^2 + 130*n + 93)*x^2 + (36*n^2 + 112*n + 46)*x^3 + (16*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(20*n + (20*n^2 + 14*n)*x + (8*n^3 + 4*n^2)*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] = ((80*n - 20 + (100*n^2 + 126*n - 44)*x + (32*n^3 + 128*n^2 + 40*n - 28)*x^2 + (24*n^3 + 36*n^2 - 8*n - 4)*x^3)*v[i - 1] - (25 + (50*n + 80)*x + (20*n^2 + 130*n + 93)*x^2 + (36*n^2 + 112*n + 46)*x^3 + (16*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(20*n + (20*n^2 + 14*n)*x + (8*n^3 + 4*n^2)*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-79ebafb266da3b41","steps":["basis A007318","\u03b8+1"],"edges":["e-1a8887aa0d003bc9","e-75b00490d3f929c1"],"text":"A172431 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A172431 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}