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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":"ef0f801211439207bed1870fff7f2443bab236ebf140d4c1ff1838eac98fe1ce","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-6120ac587df8130e","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).","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,    2;\n  2:   4,   10;\n  3:   8,   34,     6;\n  4:  16,   98,    48;\n  5:  32,  258,   234,    16;\n  6:  64,  642,   900,   176;\n  7: 128, 1538,  3006,  1120,   40;\n  8: 256, 3586,  9144,  5408,  560;\n  9: 512, 8194, 26034, 22000, 4400, 96;\n","recurrence_sentence":"The row polynomials satisfy (1-5*n*x-(n^2+16*n+3)*x^2+(3*n^3+10*n^2+9*n+2)*x^3)*P(n,x) = (2-8*n*x-(3*n^2+41*n+6)*x^2+(5*n^3+19*n^2-20*n+4)*x^3+(4*n^3+20*n^2+24*n)*x^4)*P(n-1,x) - (x+12*x^2-(n^2+5*n-21)*x^3-(7*n^2+35*n+82)*x^4+(8*n^2+40*n+48)*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[((2 - 8*n*x - (3*n^2 + 41*n + 6)*x^2 + (5*n^3 + 19*n^2 - 20*n + 4)*x^3 + (4*n^3 + 20*n^2 + 24*n)*x^4)*P[n - 1, x] - (x + 12*x^2 - (n^2 + 5*n - 21)*x^3 - (7*n^2 + 35*n + 82)*x^4 + (8*n^2 + 40*n + 48)*x^5)*D[P[n - 1, x], x])/(1 - 5*n*x - (n^2 + 16*n + 3)*x^2 + (3*n^3 + 10*n^2 + 9*n + 2)*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] = ((2 - 8*n*x - (3*n^2 + 41*n + 6)*x^2 + (5*n^3 + 19*n^2 - 20*n + 4)*x^3 + (4*n^3 + 20*n^2 + 24*n)*x^4)*v[i - 1] - (x + 12*x^2 - (n^2 + 5*n - 21)*x^3 - (7*n^2 + 35*n + 82)*x^4 + (8*n^2 + 40*n + 48)*x^5)*deriv(v[i - 1], x))/(1 - 5*n*x - (n^2 + 16*n + 3)*x^2 + (3*n^3 + 10*n^2 + 9*n + 2)*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-6120ac587df8130e","steps":["\u03b8+1"],"edges":["e-c0c95cecfdfdf639"],"text":"A119808 \u2192 \u03b8+1"},"headline":"A119808 \u2192 \u03b8+1","chain":null}