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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":"1361894992f31692ee20e657ef6a87944a8080f1cc61c41ed4877ab4a4c66ee2","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-1bffaa6198344f93","role":"sequence","name":"Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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,     8;\n  2:   4,    40,     36;\n  3:   8,   128,    288,     128;\n  4:  16,   352,   1332,    1408,     400;\n  5:  32,   896,   4824,    8576,    5600,    1152;\n  6:  64,  2176,  15264,   39040,   42400,   19584,    3136;\n  7: 128,  5120,  44352,  149504,  233600,  177408,   62720,    8192;\n  8: 256, 11776, 121536,  510976, 1056400, 1149696,  661696,  188416,  20736;\n  9: 512, 26624, 319104, 1611776, 4176800, 6014592, 4936064, 2269184, 539136, 51200;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^2+4*n+(18*n^3-8*n^2-24*n)*x+(9*n^4-21*n^3-60*n^2)*x^2-(9*n^4+39*n^3+16*n^2+16*n)*x^3+(6*n^3+8*n^2)*x^4)*P(n,x) = -(8*n^2+8*n+(36*n^3+80*n^2-16*n-64)*x+(18*n^4+138*n^3-308*n-104)*x^2+(36*n^4+48*n^3-284*n^2-392*n-144)*x^3-(78*n^3+242*n^2+212*n+80)*x^4+(24*n^2+56*n+32)*x^5)*P(n-1,x) + ((8*n+8)*x+(36*n^2+96*n+48)*x^2+(18*n^3+210*n^2+316*n+88)*x^3+(72*n^3+288*n^2+244*n+64)*x^4+(18*n^3+18*n^2-60*n-48)*x^5-(12*n^2+28*n+16)*x^6)*d/dx 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[(-(8*n^2 + 8*n + (36*n^3 + 80*n^2 - 16*n - 64)*x + (18*n^4 + 138*n^3 - 308*n - 104)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (78*n^3 + 242*n^2 + 212*n + 80)*x^4 + (24*n^2 + 56*n + 32)*x^5)*P[n - 1, x] + ((8*n + 8)*x + (36*n^2 + 96*n + 48)*x^2 + (18*n^3 + 210*n^2 + 316*n + 88)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (18*n^3 + 18*n^2 - 60*n - 48)*x^5 - (12*n^2 + 28*n + 16)*x^6)*D[P[n - 1, x], x])/(-4*n^2 - 4*n - (18*n^3 - 8*n^2 - 24*n)*x - (9*n^4 - 21*n^3 - 60*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (6*n^3 + 8*n^2)*x^4)]];\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] = (-(8*n^2 + 8*n + (36*n^3 + 80*n^2 - 16*n - 64)*x + (18*n^4 + 138*n^3 - 308*n - 104)*x^2 + (36*n^4 + 48*n^3 - 284*n^2 - 392*n - 144)*x^3 - (78*n^3 + 242*n^2 + 212*n + 80)*x^4 + (24*n^2 + 56*n + 32)*x^5)*v[i - 1] + ((8*n + 8)*x + (36*n^2 + 96*n + 48)*x^2 + (18*n^3 + 210*n^2 + 316*n + 88)*x^3 + (72*n^3 + 288*n^2 + 244*n + 64)*x^4 + (18*n^3 + 18*n^2 - 60*n - 48)*x^5 - (12*n^2 + 28*n + 16)*x^6)*deriv(v[i - 1], x))/(-4*n^2 - 4*n - (18*n^3 - 8*n^2 - 24*n)*x - (9*n^4 - 21*n^3 - 60*n^2)*x^2 + (9*n^4 + 39*n^3 + 16*n^2 + 16*n)*x^3 - (6*n^3 + 8*n^2)*x^4)); 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-1bffaa6198344f93","steps":["(\u03b8+1)\u00b2"],"edges":["e-a6e970684438cd5f"],"text":"A059473 \u2192 (\u03b8+1)\u00b2"},"headline":"A059473 \u2192 (\u03b8+1)\u00b2","chain":null}