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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":"a72a27eb42359c2acb9771a518807ecaf9adcfc7372734ca762c28d92b873876","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-be7df2e3fd154f3c","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:        15,         176,          288;\n  3:       105,        1952,         6768,          6144;\n  4:       945,       24288,       131184,        247808,         153600;\n  5:     10395,      342768,      2535984,       7390208,        9433600,        4423680;\n  6:    135135,     5453904,     51477264,     204274688,      399769600,      383533056,      144506880;\n  7:   2027025,    96943680,   1115754912,    5626544128,    14933561600,    21871706112,    16784474112,     5284823040;\n  8:  34459425,  1907501760,  25947778464,  159125307392,   535197753600,  1059957522432,  1237970370560,   791364501504,   214035333120;\n  9: 654729075, 41204690640, 647515433376, 4684038662144, 19119555859200, 47883844472832, 75459355639808, 73245797646336, 40123216429056, 9512681472000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(32*n^3-48*n^2+16*n)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((32*n^3-48*n^2+16*n)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(16*n^2-16*n)*x)*P(n-1,x) - (16*n^2-16*n)*x^2*d/dx 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[(-(4*n^3 - 4*n^2 + n + (32*n^3 - 48*n^2 + 16*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((32*n^3 - 48*n^2 + 16*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (16*n^2 - 16*n)*x)*P[n - 1, x] - (16*n^2 - 16*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\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] = (-(4*n^3 - 4*n^2 + n + (32*n^3 - 48*n^2 + 16*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((32*n^3 - 48*n^2 + 16*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (16*n^2 - 16*n)*x)*v[i - 1] - (16*n^2 - 16*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-7fd16a32c438d522","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-5d9e378754634336","e-68bc3324f16df41c"],"text":"A059364 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A059364 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}