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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":"8402030d50ff5086c06fa0e67d62aa885a7d1f3af4f91a3dbc631a7afe809b05","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-e4d248920a317259","role":"sequence","name":"Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","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: 0, 4, 18;\n  2: 0, 0,  9, 96, 150;\n  3: 0, 0,  0, 16, 300, 1080,  980;\n  4: 0, 0,  0,  0,  25,  720, 4410,  8960,  5670;\n  5: 0, 0,  0,  0,   0,   36, 1470, 13440, 45360,  63000,  30492;\n  6: 0, 0,  0,  0,   0,    0,   49,  2688, 34020, 168000, 381150,  399168,  156156;\n  7: 0, 0,  0,  0,   0,    0,    0,    64,  4536,  75600, 508200, 1663200, 2810808,  2354352,   772200;\n  8: 0, 0,  0,  0,   0,    0,    0,     0,    81,   7200, 152460, 1330560, 5855850, 14126112, 18918900,  13178880,   3719430;\n  9: 0, 0,  0,  0,   0,    0,    0,     0,     0,    100,  10890,  285120, 3123120, 17657640, 56756700, 107627520, 119021760, 70887960, 17551820;\n","recurrence_sentence":"The row polynomials satisfy (n^4-(2*n^3-n^2)*x)*P(n,x) = -((n^4-2*n^3-3*n^2)*x-(8*n^3+6*n^2)*x^2+(8*n^2+8*n+2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2)*x^2+(2*n^3-2*n^2-4*n-1)*x^3-(4*n^2+4*n+1)*x^4)*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[(-((n^4 - 2*n^3 - 3*n^2)*x - (8*n^3 + 6*n^2)*x^2 + (8*n^2 + 8*n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2)*x^2 + (2*n^3 - 2*n^2 - 4*n - 1)*x^3 - (4*n^2 + 4*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - (2*n^3 - n^2)*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] = (-((n^4 - 2*n^3 - 3*n^2)*x - (8*n^3 + 6*n^2)*x^2 + (8*n^2 + 8*n + 2)*x^3)*v[i - 1] + ((2*n^3 + 2*n^2)*x^2 + (2*n^3 - 2*n^2 - 4*n - 1)*x^3 - (4*n^2 + 4*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 - (2*n^3 - n^2)*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-e4d248920a317259","steps":["(\u03b8+1)\u00b2"],"edges":["e-24889bda8a9a25db"],"text":"A109983 \u2192 (\u03b8+1)\u00b2"},"headline":"A109983 \u2192 (\u03b8+1)\u00b2","chain":null}