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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":"72b92e944b0a3e97890e3f4424ec1f335da4b07aaa99d3611c952125d604dd4b","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-a313e9d783515c70","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+3*x)^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:      4,       36;\n  2:     16,      288,       729;\n  3:     64,     1728,      8748,      11664;\n  4:    256,     9216,     69984,     186624,      164025;\n  5:   1024,    46080,    466560,    1866240,     3280500,     2125764;\n  6:   4096,   221184,   2799360,   14929920,    39366000,    51018336,     26040609;\n  7:  16384,  1032192,  15676416,  104509440,   367416000,   714256704,    729137052,    306110016;\n  8:  65536,  4718592,  83607552,  668860416,  2939328000,  7618738176,  11666192832,   9795520512,   3486784401;\n  9: 262144, 21233664, 429981696, 4013162496, 21163161600, 68568643584, 139994313984, 176319369216, 125524238436, 38742048900;\n","recurrence_sentence":"The row polynomials satisfy (16+(108*n-36)*x+81*n^2*x^2)*P(n,x) = (64+(432*n+432)*x+(324*n^2+1620*n+972)*x^2+(729*n^2+1458*n+729)*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[((64 + (432*n + 432)*x + (324*n^2 + 1620*n + 972)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*P[n - 1, x])/(16 + (108*n - 36)*x + 81*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] = ((64 + (432*n + 432)*x + (324*n^2 + 1620*n + 972)*x^2 + (729*n^2 + 1458*n + 729)*x^3)*v[i - 1])/(16 + (108*n - 36)*x + 81*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-6a5af5e1168bb679","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-39440d1ff583ab52","e-16c481888ee0b130"],"text":"A013610 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A013610 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}