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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":"4d494fbecdf25a968c2fb27c93b968bcdfa7d370696311e07e81046369004f68","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-2872b2f2cd7c5725","role":"sequence","name":"Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, 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: 1,    4;\n  2: 1,   20,      9;\n  3: 1,   56,    126,       16;\n  4: 1,  132,    801,      528,       25;\n  5: 1,  288,   3717,     6608,     1800,       36;\n  6: 1,  604,  14688,    54288,    40800,     5436,       49;\n  7: 1, 1240,  52866,   354320,   553625,   211464,    15190,      64;\n  8: 1, 2516, 179487,  2007296,  5621075,  4516416,   977207,   40256,     81;\n  9: 1, 5072, 586539, 10356784, 47477400, 68367456, 31717651, 4170944, 102708, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 11*x^2*P(n-4,x) + 5*x^3*d/dx P(n-4,x) - (4*x-(20*n-15)*x^2)*P(n-3,x) + (5*x^2+(9*n-23)*x^3)*d/dx P(n-3,x) + (5*x^3-5*x^4)*d^2/dx^2 P(n-3,x) - ((4*n-8)*x-(9*n^2-2*n-7)*x^2)*P(n-2,x) - (2*x-(4*n+4)*x^2-(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) - (x^2-(4*n+4)*x^3+(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (1+4*n*x)*P(n-1,x) + (2*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 20*x + 9*x^2, P(3,x) = 1 + 56*x + 126*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 20*x + 9*x^2;\nP[3, x_] := 1 + 56*x + 126*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[11*x^2*P[n - 4, x] + 5*x^3*D[P[n - 4, x], x] - (4*x - (20*n - 15)*x^2)*P[n - 3, x] + (5*x^2 + (9*n - 23)*x^3)*D[P[n - 3, x], x] + (5*x^3 - 5*x^4)*D[P[n - 3, x], {x, 2}] - ((4*n - 8)*x - (9*n^2 - 2*n - 7)*x^2)*P[n - 2, x] - (2*x - (4*n + 4)*x^2 - (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 4*n*x)*P[n - 1, x] + (2*x - (4*n + 4)*x^2)*D[P[n - 1, x], x]];\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; v[3] = 1 + 20*x + 9*x^2; v[4] = 1 + 56*x + 126*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = 11*x^2*v[i - 4] + 5*x^3*deriv(v[i - 4], x) - (4*x - (20*n - 15)*x^2)*v[i - 3] + (5*x^2 + (9*n - 23)*x^3)*deriv(v[i - 3], x) + (5*x^3 - 5*x^4)*deriv(deriv(v[i - 3], x), x) - ((4*n - 8)*x - (9*n^2 - 2*n - 7)*x^2)*v[i - 2] - (2*x - (4*n + 4)*x^2 - (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) - (x^2 - (4*n + 4)*x^3 + (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 4*n*x)*v[i - 1] + (2*x - (4*n + 4)*x^2)*deriv(v[i - 1], x)); 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-2872b2f2cd7c5725","steps":["(\u03b8+1)\u00b2"],"edges":["e-55abd557d63ba53c"],"text":"A144438 \u2192 (\u03b8+1)\u00b2"},"headline":"A144438 \u2192 (\u03b8+1)\u00b2","chain":null}