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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":"91c89c1af569635bfdb38452bc96179365a91fc29e7a2eb00f6714c245642bc4","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-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:    9,    24,      9;\n  1:   18,    84,     72,      16;\n  2:   36,   240,    333,     160,      25;\n  3:   72,   624,   1206,     912,     300,      36;\n  4:  144,  1536,   3816,    3968,    2025,     504,      49;\n  5:  288,  3648,  11088,   14720,   10250,    3924,     784,     64;\n  6:  576,  8448,  30384,   49152,   43500,   22608,    6909,   1152,     81;\n  7: 1152, 19200,  79776,  152320,  163800,  107856,   44590,  11328,   1620,   100;\n  8: 2304, 43008, 202752,  446464,  565600,  451584,  235984,  80896,  17577,  2200,  121;\n  9: 4608, 95232, 502272, 1253376, 1828800, 1717632, 1086624, 470016, 137538, 26100, 2904, 144;\n","recurrence_sentence":"The row polynomials satisfy (36+(54*n+78)*x+(9*n^2+60*n+72)*x^2+(6*n^2+26*n+28)*x^3+(n^2+4*n+4)*x^4)*P(n,x) = (72+(108*n+300)*x+(18*n^2+210*n+414)*x^2+(21*n^2+154*n+261)*x^3+(8*n^2+50*n+78)*x^4+(n^2+6*n+9)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 24*x + 9*x^2.","mathematica":"P[0, x_] := 9 + 24*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((72 + (108*n + 300)*x + (18*n^2 + 210*n + 414)*x^2 + (21*n^2 + 154*n + 261)*x^3 + (8*n^2 + 50*n + 78)*x^4 + (n^2 + 6*n + 9)*x^5)*P[n - 1, x])/(36 + (54*n + 78)*x + (9*n^2 + 60*n + 72)*x^2 + (6*n^2 + 26*n + 28)*x^3 + (n^2 + 4*n + 4)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 9 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (108*n + 300)*x + (18*n^2 + 210*n + 414)*x^2 + (21*n^2 + 154*n + 261)*x^3 + (8*n^2 + 50*n + 78)*x^4 + (n^2 + 6*n + 9)*x^5)*v[i - 1])/(36 + (54*n + 78)*x + (9*n^2 + 60*n + 72)*x^2 + (6*n^2 + 26*n + 28)*x^3 + (n^2 + 4*n + 4)*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-f48bed27dbc29c5f","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-3fc4d53dea6f488e","e-3a8b443f258ded85"],"text":"A391533 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A391533 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}