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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":"2b4a13d7f54b93dec4ea6b198cfafad7293f97e4e24cb138caab7ec004b3c309","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-b4785bc61afd940f","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:    3,     16;\n  2:    8,     72,      72;\n  3:   21,    264,     540,     256;\n  4:   55,    888,    2736,    2688,      800;\n  5:  144,   2840,   11664,   17536,    10800,     2304;\n  6:  377,   8784,   45036,   91392,    86400,    38016,     6272;\n  7:  987,  26528,  162972,  416768,   534000,   360576,   122304,   16384;\n  8: 2584,  78696,  563256, 1738368,  2818000,  2581632,  1342208,  368640,   41472;\n  9: 6765, 230232, 1880928, 6801024, 13352400, 15500160, 10932096, 4595712, 1057536, 102400;\n","recurrence_sentence":"The row polynomials satisfy (25+580*x-(140*n^2-840*n-2336)*x^2-(1040*n^2-3840*n-1808)*x^3+(16*n^4-192*n^3-1056*n^2+4320*n-1296)*x^4+(64*n^4-384*n^3+576*n^2)*x^5)*P(n,x) = -(25+580*x-(140*n^2-560*n-3036)*x^2-(1040*n^2-1760*n-4608)*x^3+(16*n^4-128*n^3-1536*n^2+1696*n+1792)*x^4+(64*n^4-128*n^3-192*n^2+256*n+256)*x^5)*P(n-2,x) + (75+1940*x-(420*n^2-2100*n-12068)*x^2-(3520*n^2-10400*n-27872)*x^3+(48*n^4-480*n^3-6976*n^2+17744*n+26144)*x^4+(224*n^4-1088*n^3-4000*n^2+10752*n+8832)*x^5+(128*n^4-512*n^3-256*n^2+1536*n+1152)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(25 + 580*x - (140*n^2 - 560*n - 3036)*x^2 - (1040*n^2 - 1760*n - 4608)*x^3 + (16*n^4 - 128*n^3 - 1536*n^2 + 1696*n + 1792)*x^4 + (64*n^4 - 128*n^3 - 192*n^2 + 256*n + 256)*x^5)*P[n - 2, x] + (75 + 1940*x - (420*n^2 - 2100*n - 12068)*x^2 - (3520*n^2 - 10400*n - 27872)*x^3 + (48*n^4 - 480*n^3 - 6976*n^2 + 17744*n + 26144)*x^4 + (224*n^4 - 1088*n^3 - 4000*n^2 + 10752*n + 8832)*x^5 + (128*n^4 - 512*n^3 - 256*n^2 + 1536*n + 1152)*x^6)*P[n - 1, x])/(25 + 580*x - (140*n^2 - 840*n - 2336)*x^2 - (1040*n^2 - 3840*n - 1808)*x^3 + (16*n^4 - 192*n^3 - 1056*n^2 + 4320*n - 1296)*x^4 + (64*n^4 - 384*n^3 + 576*n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + 580*x - (140*n^2 - 560*n - 3036)*x^2 - (1040*n^2 - 1760*n - 4608)*x^3 + (16*n^4 - 128*n^3 - 1536*n^2 + 1696*n + 1792)*x^4 + (64*n^4 - 128*n^3 - 192*n^2 + 256*n + 256)*x^5)*v[i - 2] + (75 + 1940*x - (420*n^2 - 2100*n - 12068)*x^2 - (3520*n^2 - 10400*n - 27872)*x^3 + (48*n^4 - 480*n^3 - 6976*n^2 + 17744*n + 26144)*x^4 + (224*n^4 - 1088*n^3 - 4000*n^2 + 10752*n + 8832)*x^5 + (128*n^4 - 512*n^3 - 256*n^2 + 1536*n + 1152)*x^6)*v[i - 1])/(25 + 580*x - (140*n^2 - 840*n - 2336)*x^2 - (1040*n^2 - 3840*n - 1808)*x^3 + (16*n^4 - 192*n^3 - 1056*n^2 + 4320*n - 1296)*x^4 + (64*n^4 - 384*n^3 + 576*n^2)*x^5)); 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-353fb5a029e4e669","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-3748e9c6bfc37784","e-b79ceb6f18835f10"],"text":"A207607 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A207607 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}