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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":"7e2836800bcfb2b657eea4c5277192ccfb46026bb6a44649566df9cdfe27130d","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-8ac1223942f7b8c5","role":"sequence","name":"reversal of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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,  8;\n  2: 1, 16,   45;\n  3: 1, 24,  138,   224;\n  4: 1, 32,  279,   928,   1045;\n  5: 1, 40,  468,  2368,   5455,   4680;\n  6: 1, 48,  705,  4800,  16830,  29472,   20377;\n  7: 1, 56,  990,  8480,  40050, 106704,  150276,   86912;\n  8: 1, 64, 1323, 13664,  81275, 291744,  624666,  734592,  364905;\n  9: 1, 72, 1704, 20608, 147945, 669144, 1930068, 3447936, 3477357, 1513160;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(16*n^3+8*n^2)*x+(24*n^3-24*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(16*n^3+56*n^2+8*n-8)*x+(120*n^3+192*n^2-68*n-52)*x^2+(240*n^3+144*n^2-288*n-96)*x^3+(144*n^3-144*n)*x^4)*P(n-1,x) - ((2*n+1)*x+(8*n^2+32*n+12)*x^2+(76*n^2+164*n+44)*x^3+(192*n^2+288*n+48)*x^4+(144*n^2+144*n)*x^5)*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[((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (120*n^3 + 192*n^2 - 68*n - 52)*x^2 + (240*n^3 + 144*n^2 - 288*n - 96)*x^3 + (144*n^3 - 144*n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (76*n^2 + 164*n + 44)*x^3 + (192*n^2 + 288*n + 48)*x^4 + (144*n^2 + 144*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (24*n^3 - 24*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] = ((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (120*n^3 + 192*n^2 - 68*n - 52)*x^2 + (240*n^3 + 144*n^2 - 288*n - 96)*x^3 + (144*n^3 - 144*n)*x^4)*v[i - 1] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (76*n^2 + 164*n + 44)*x^3 + (192*n^2 + 288*n + 48)*x^4 + (144*n^2 + 144*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (24*n^3 - 24*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-30cb3cad216ed189","steps":["reverse","\u03b8+1"],"edges":["e-73dd335e950d1f81","e-d80b429919e51d6c"],"text":"A207823 \u2192 reverse \u2192 \u03b8+1"},"headline":"A207823 \u2192 reverse \u2192 \u03b8+1","chain":null}