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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":"5402e63c9889423d44e20de1934e3ecd0eeffa91703ea2b94d8738721a59890c","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-b4814297dd192d81","role":"sequence","name":"Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","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":[{"id":"l-113849d64f19cdaa","kind":"oeis_match","node_a_id":"n-ba78222af553c074","node_b_id":null,"fingerprint":null,"oeis_id":"A034428","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034428","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A034428","relation":"match_row_sums"}],"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;\n  2: 1,    2;\n  3: 1,    8;\n  4: 1,   22,    12;\n  5: 1,   52,   102;\n  6: 1,  114,   540,    136;\n  7: 1,  240,  2304,   1984;\n  8: 1,  494,  8712,  17152,  2480;\n  9: 1, 1004, 30582, 115072, 55280;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-n-2)*x^2*P(n-2,x) - ((n+1)*x^2-(2*n+2)*x^3)*d/dx P(n-2,x) + (1+(2*n-2)*x)*P(n-1,x) + (x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - n - 2)*x^2*P[n - 2, x] - ((n + 1)*x^2 - (2*n + 2)*x^3)*D[P[n - 2, x], x] + (1 + (2*n - 2)*x)*P[n - 1, x] + (x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - n - 2)*x^2*v[i - 2] - ((n + 1)*x^2 - (2*n + 2)*x^3)*deriv(v[i - 2], x) + (1 + (2*n - 2)*x)*v[i - 1] + (x - 2*x^2)*deriv(v[i - 1], 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-ff270bcb802a85af","steps":["\u03b3 (scale 1/2) (shift +1)","\u03b8+1"],"edges":["e-ea229ed6e1820773","e-1ae338480129f9a5"],"text":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 \u03b8+1"},"headline":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 \u03b8+1","chain":null}