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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":"22a1a065b86d3bf85c74b6929d977bbe5df8bc36312a0534ce9e285085a84aa7","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-5733d79f45a14084","role":"sequence","name":"Triangle of coefficients in expansion of (2+3x)^n.","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,    12;\n  2:   4,    48,     81;\n  3:   8,   144,    486,     432;\n  4:  16,   384,   1944,    3456,    2025;\n  5:  32,   960,   6480,   17280,   20250,     8748;\n  6:  64,  2304,  19440,   69120,  121500,   104976,    35721;\n  7: 128,  5376,  54432,  241920,  567000,   734832,   500094,   139968;\n  8: 256, 12288, 145152,  774144, 2268000,  3919104,  4000752,  2239488,  531441;\n  9: 512, 27648, 373248, 2322432, 8164800, 17635968, 24004512, 20155392, 9565938, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (8+(36*n+36)*x+(18*n^2+90*n+54)*x^2+(27*n^2+54*n+27)*x^3)*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[((8 + (36*n + 36)*x + (18*n^2 + 90*n + 54)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*n^2*x^2)]];\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] = ((8 + (36*n + 36)*x + (18*n^2 + 90*n + 54)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*n^2*x^2)); 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-5733d79f45a14084","steps":["(\u03b8+1)\u00b2"],"edges":["e-e6c15cbf1d46e331"],"text":"A013620 \u2192 (\u03b8+1)\u00b2"},"headline":"A013620 \u2192 (\u03b8+1)\u00b2","chain":null}