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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":"afd89ac59a1bb0e28f9ed7bc1f331a3090aced57d45da80371d32b009fd743e2","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-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 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,       8;\n  2:     9,      72,       48;\n  3:    28,     336,      768,       256;\n  4:   101,    1616,     5952,      5888,      1280;\n  5:   342,    6840,    35520,     65280,     38400,      6144;\n  6:  1189,   28536,   190320,    508160,    556800,    227328,     28672;\n  7:  4088,  114464,   934656,   3304960,   5478400,   4079616,   1261568,    131072;\n  8: 14121,  451872,  4363776,  19210752,  42969600,  48955392,  27009024,   6684672,   589824;\n  9: 48682, 1752552, 19545984, 103143936, 292008960, 456044544, 383975424, 166330368, 34209792, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (6*n*x+(12*n^2+24*n)*x^2+(8*n^3+24*n^2+16*n)*x^3)*P(n,x) = -(12*n-12+(36*n^2+66*n-114)*x+(24*n^3+180*n^2+20*n-344)*x^2+(88*n^3+144*n^2-264*n-352)*x^3-(64*n^2+192*n+128)*x^4)*P(n-1,x) + (3+(12*n+42)*x+(12*n^2+132*n+218)*x^2+(96*n^2+488*n+500)*x^3+(200*n^2+632*n+448)*x^4+(32*n^2+96*n+64)*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[(-(12*n - 12 + (36*n^2 + 66*n - 114)*x + (24*n^3 + 180*n^2 + 20*n - 344)*x^2 + (88*n^3 + 144*n^2 - 264*n - 352)*x^3 - (64*n^2 + 192*n + 128)*x^4)*P[n - 1, x] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 218)*x^2 + (96*n^2 + 488*n + 500)*x^3 + (200*n^2 + 632*n + 448)*x^4 + (32*n^2 + 96*n + 64)*x^5)*D[P[n - 1, x], x])/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*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] = (-(12*n - 12 + (36*n^2 + 66*n - 114)*x + (24*n^3 + 180*n^2 + 20*n - 344)*x^2 + (88*n^3 + 144*n^2 - 264*n - 352)*x^3 - (64*n^2 + 192*n + 128)*x^4)*v[i - 1] + (3 + (12*n + 42)*x + (12*n^2 + 132*n + 218)*x^2 + (96*n^2 + 488*n + 500)*x^3 + (200*n^2 + 632*n + 448)*x^4 + (32*n^2 + 96*n + 64)*x^5)*deriv(v[i - 1], x))/(6*n*x + (12*n^2 + 24*n)*x^2 + (8*n^3 + 24*n^2 + 16*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 A013611","\u03b8+1"],"edges":["e-09b3cf169dcfe004","e-7f144882926099ed"],"text":"A081580 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A081580 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}