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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":"48d52043145230eba4e7679aadb6fc8c47de57db057dc55f42e23993f0479318","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-4f8c7533ab8b37bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,5,1) array.","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:     2,      2;\n  2:     9,     18,       3;\n  3:    28,     84,      48,       4;\n  4:   101,    404,     372,      92,       5;\n  5:   342,   1710,    2220,    1020,     150,      6;\n  6:  1189,   7134,   11895,    7940,    2175,    222,     7;\n  7:  4088,  28616,   58416,   51640,   21400,   3984,   308,     8;\n  8: 14121, 112968,  272736,  300168,  167850,  47808,  6594,   408,   9;\n  9: 48682, 438138, 1221624, 1611624, 1140660, 445356, 93744, 10152, 522, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(96*n-96+(72*n^2+132*n-228)*x+(12*n^3+90*n^2+10*n-172)*x^2+(11*n^3+18*n^2-33*n-44)*x^3-(2*n^2+6*n+4)*x^4)*P(n-1,x) + (96+(96*n+336)*x+(24*n^2+264*n+436)*x^2+(48*n^2+244*n+250)*x^3+(25*n^2+79*n+56)*x^4+(n^2+3*n+2)*x^5)*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[(-(96*n - 96 + (72*n^2 + 132*n - 228)*x + (12*n^3 + 90*n^2 + 10*n - 172)*x^2 + (11*n^3 + 18*n^2 - 33*n - 44)*x^3 - (2*n^2 + 6*n + 4)*x^4)*P[n - 1, x] + (96 + (96*n + 336)*x + (24*n^2 + 264*n + 436)*x^2 + (48*n^2 + 244*n + 250)*x^3 + (25*n^2 + 79*n + 56)*x^4 + (n^2 + 3*n + 2)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];\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] = (-(96*n - 96 + (72*n^2 + 132*n - 228)*x + (12*n^3 + 90*n^2 + 10*n - 172)*x^2 + (11*n^3 + 18*n^2 - 33*n - 44)*x^3 - (2*n^2 + 6*n + 4)*x^4)*v[i - 1] + (96 + (96*n + 336)*x + (24*n^2 + 264*n + 436)*x^2 + (48*n^2 + 244*n + 250)*x^3 + (25*n^2 + 79*n + 56)*x^4 + (n^2 + 3*n + 2)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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-6ad01dc6e705cdb6","steps":["basis A007318","\u03b8+1"],"edges":["e-1ae1e4b2be227437","e-72549e5347e82298"],"text":"A081580 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A081580 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}