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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":"92e6d825d8dde3c5feffe42ab899892512959bd8b3cf824a7c57a4c55aef56c4","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-01cea33b6485d9cc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","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:     9,     24,     12;\n  3:    27,    108,    108,      32;\n  4:    81,    432,    648,     384,      80;\n  5:   243,   1620,   3240,    2880,    1200,     192;\n  6:   729,   5832,  14580,   17280,   10800,    3456,     448;\n  7:  2187,  20412,  61236,   90720,   75600,   36288,    9408,   1024;\n  8:  6561,  69984, 244944,  435456,  453600,  290304,  112896,  24576,  2304;\n  9: 19683, 236196, 944784, 1959552, 2449440, 1959552, 1016064, 331776, 62208, 5120;\n","recurrence_sentence":"The row polynomials satisfy (3+2*n*x)*P(n,x) = (9+(6*n+12)*x+(4*n+4)*x^2)*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 + (6*n + 12)*x + (4*n + 4)*x^2)*P[n - 1, x])/(3 + 2*n*x)]];\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 + (6*n + 12)*x + (4*n + 4)*x^2)*v[i - 1])/(3 + 2*n*x)); 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-01cea33b6485d9cc","steps":["\u03b8+1"],"edges":["e-28778a02b676d631"],"text":"A038220 \u2192 \u03b8+1"},"headline":"A038220 \u2192 \u03b8+1","chain":null}