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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":"da5178df79fc84b5dcc90db277de8d53d297cbb5b1916fee20ed89a5f6e5825d","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-b0d4d18dfd70fd8a","role":"sequence","name":"basis_map_A013610 of Even row Pascal-square read by antidiagonals.","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,     12;\n  2:    8,     60,      81;\n  3:   21,    228,     594,      432;\n  4:   55,    780,    2997,     4320,     2025;\n  5:  144,   2520,   12798,    27648,    26325,     8748;\n  6:  377,   7848,   49572,   142992,   206955,   142884,    35721;\n  7:  987,  23820,  179982,   650160,  1267650,  1347192,   714420,   139968;\n  8: 2584,  70932,  623943,  2709504,  6656175,  9605304,  7930062,  3359232,   531441;\n  9: 6765, 208140, 2089206, 10601280, 31446225, 57488940, 64910160, 43250112, 15057495, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (60*n+(180*n^2+126*n)*x+(216*n^3+108*n^2)*x^2)*P(n,x) = (240*n-60+(900*n^2+1134*n-396)*x+(864*n^3+3456*n^2+1080*n-756)*x^2+(1944*n^3+2916*n^2-648*n-324)*x^3)*P(n-1,x) - (25+(150*n+240)*x+(180*n^2+1170*n+837)*x^2+(972*n^2+3024*n+1242)*x^3+(1296*n^2+2592*n+648)*x^4)*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[((240*n - 60 + (900*n^2 + 1134*n - 396)*x + (864*n^3 + 3456*n^2 + 1080*n - 756)*x^2 + (1944*n^3 + 2916*n^2 - 648*n - 324)*x^3)*P[n - 1, x] - (25 + (150*n + 240)*x + (180*n^2 + 1170*n + 837)*x^2 + (972*n^2 + 3024*n + 1242)*x^3 + (1296*n^2 + 2592*n + 648)*x^4)*D[P[n - 1, x], x])/(60*n + (180*n^2 + 126*n)*x + (216*n^3 + 108*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] = ((240*n - 60 + (900*n^2 + 1134*n - 396)*x + (864*n^3 + 3456*n^2 + 1080*n - 756)*x^2 + (1944*n^3 + 2916*n^2 - 648*n - 324)*x^3)*v[i - 1] - (25 + (150*n + 240)*x + (180*n^2 + 1170*n + 837)*x^2 + (972*n^2 + 3024*n + 1242)*x^3 + (1296*n^2 + 2592*n + 648)*x^4)*deriv(v[i - 1], x))/(60*n + (180*n^2 + 126*n)*x + (216*n^3 + 108*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-79ebafb266da3b41","steps":["basis A013610","\u03b8+1"],"edges":["e-01dc3c5cce6c6f01","e-0f5fe450ad89f5c8"],"text":"A172431 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A172431 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}