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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":"765c0c84b6538ebf3c33ae541967053ae9404f550fb8cc29fb3f73de073361d2","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-ecd77b030b8fab94","role":"sequence","name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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: 1,   4;\n  2: 1,  16,     27;\n  3: 1,  40,    243,     288;\n  4: 1,  80,   1143,    4608,      4500;\n  5: 1, 140,   3843,   35088,    112500,      97200;\n  6: 1, 224,  10458,  178560,   1263825,    3499200,     2778300;\n  7: 1, 336,  24570,  699136,   9075825,   54456624,   136136700,    101606400;\n  8: 1, 480,  51786, 2271616,  48402225,  524947392,  2804511276,   6502809600,   4629441600;\n  9: 1, 660, 100386, 6414496, 208125225, 3661411572, 34957539036, 171339390720, 374984769600, 257191200000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(8*n^3-8*n)*x-(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + ((8*n^3-8*n)*x^2+(4*n^5-4*n^4-4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (8*n+(8*n^3-8*n)*x)*P(n-1,x) - (8*n^3-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (8*n^3 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((8*n^3 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (8*n + (8*n^3 - 8*n)*x)*P[n - 1, x] - (8*n^3 - 8*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (8*n^3 - 8*n)*x - (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((8*n^3 - 8*n)*x^2 + (4*n^5 - 4*n^4 - 4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (8*n + (8*n^3 - 8*n)*x)*v[i - 1] - (8*n^3 - 8*n)*x^2*deriv(v[i - 1], x))/(4*n - 4)); 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-ecd77b030b8fab94","steps":["(\u03b8+1)\u00b2"],"edges":["e-a3debe476fcbae67"],"text":"A128813 \u2192 (\u03b8+1)\u00b2"},"headline":"A128813 \u2192 (\u03b8+1)\u00b2","chain":null}