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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":"061dc444d975d74dbec5c1838af56e0e8353768d7b7a7cb81b12442f6ae6f841","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-1b0c4ee1b6f9479c","role":"sequence","name":"basis_map_A013610 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-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:        1,         12;\n  2:        3,         60,         162;\n  3:       15,        408,        2025,         2592;\n  4:      105,       3576,       25191,        61344,         48600;\n  5:      945,      38484,      347409,      1223856,       1875150,       1049760;\n  6:    10395,     491364,     5380101,     24579504,      55047600,      60151248,      25719120;\n  7:   135135,    7260912,    93158262,    520390656,    1522133775,    2446605648,    2053671732,     705438720;\n  8:  2027025,  121886640,  1789463178,  11780612352,   42346656225,   89304027984,  110727527148,   74957902848,   21427701120;\n  9: 34459425, 2290991580, 37825487406, 286164642528, 1216887739425, 3164606469756, 5164290989928, 5179124489472, 2925723005424, 714256704000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(24*n^3-60*n^2+36*n)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((24*n^3-60*n^2+36*n)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (24*n^3 - 60*n^2 + 36*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((24*n^3 - 60*n^2 + 36*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (24*n^3 - 60*n^2 + 36*n)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((24*n^3 - 60*n^2 + 36*n)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-849de7332df28657","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-678ee1ecc620a9a6","e-c4f088f1c59c37e7"],"text":"A088996 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A088996 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}