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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":"4d529cd6142573f14dc54250949a6d62526528e12791c9e368f8bfb1f0d126c0","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-4f8c7533ab8b37bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,5,1) array.","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,      4;\n  2:     9,     36,       9;\n  3:    28,    168,     144,      16;\n  4:   101,    808,    1116,     368,      25;\n  5:   342,   3420,    6660,    4080,     750,      36;\n  6:  1189,  14268,   35685,   31760,   10875,    1332,     49;\n  7:  4088,  57232,  175248,  206560,  107000,   23904,   2156,    64;\n  8: 14121, 225936,  818208, 1200672,  839250,  286848,  46158,  3264,   81;\n  9: 48682, 876276, 3664872, 6446496, 5703300, 2672136, 656208, 81216, 4698, 100;\n","recurrence_sentence":"The row polynomials satisfy (96+(288*n-48)*x+(264*n^2-24*n-32)*x^2+(72*n^3+60*n^2-36*n-18)*x^3+(6*n^4+12*n^3-4*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (480+(1440*n+1680)*x+(1320*n^2+3960*n+2240)*x^2+(360*n^3+2700*n^2+4020*n+1430)*x^3+(30*n^4+540*n^3+1720*n^2+1740*n+450)*x^4+(30*n^4+175*n^3+320*n^2+215*n+50)*x^5-(5*n^3+20*n^2+25*n+10)*x^6)*P(n-2,x) + (192+(576*n+288)*x+(528*n^2+768*n+80)*x^2+(144*n^3+600*n^2+384*n-140)*x^3+(12*n^4+120*n^3+220*n^2+44*n-60)*x^4+(6*n^4+22*n^3+15*n^2+1)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((480 + (1440*n + 1680)*x + (1320*n^2 + 3960*n + 2240)*x^2 + (360*n^3 + 2700*n^2 + 4020*n + 1430)*x^3 + (30*n^4 + 540*n^3 + 1720*n^2 + 1740*n + 450)*x^4 + (30*n^4 + 175*n^3 + 320*n^2 + 215*n + 50)*x^5 - (5*n^3 + 20*n^2 + 25*n + 10)*x^6)*P[n - 2, x] + (192 + (576*n + 288)*x + (528*n^2 + 768*n + 80)*x^2 + (144*n^3 + 600*n^2 + 384*n - 140)*x^3 + (12*n^4 + 120*n^3 + 220*n^2 + 44*n - 60)*x^4 + (6*n^4 + 22*n^3 + 15*n^2 + 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(96 + (288*n - 48)*x + (264*n^2 - 24*n - 32)*x^2 + (72*n^3 + 60*n^2 - 36*n - 18)*x^3 + (6*n^4 + 12*n^3 - 4*n^2 - 4*n + 2)*x^4 - (n^3 + 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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((480 + (1440*n + 1680)*x + (1320*n^2 + 3960*n + 2240)*x^2 + (360*n^3 + 2700*n^2 + 4020*n + 1430)*x^3 + (30*n^4 + 540*n^3 + 1720*n^2 + 1740*n + 450)*x^4 + (30*n^4 + 175*n^3 + 320*n^2 + 215*n + 50)*x^5 - (5*n^3 + 20*n^2 + 25*n + 10)*x^6)*v[i - 2] + (192 + (576*n + 288)*x + (528*n^2 + 768*n + 80)*x^2 + (144*n^3 + 600*n^2 + 384*n - 140)*x^3 + (12*n^4 + 120*n^3 + 220*n^2 + 44*n - 60)*x^4 + (6*n^4 + 22*n^3 + 15*n^2 + 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(96 + (288*n - 48)*x + (264*n^2 - 24*n - 32)*x^2 + (72*n^3 + 60*n^2 - 36*n - 18)*x^3 + (6*n^4 + 12*n^3 - 4*n^2 - 4*n + 2)*x^4 - (n^3 + 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-6ad01dc6e705cdb6","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-1ae1e4b2be227437","e-3333cdcc6cafcf6b"],"text":"A081580 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A081580 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}