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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":"f2a88569167d17eee4302188150fe0a9aa8d5e52739553c4962f9941d69f8bd6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-11b174a1f5e78a68","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 0, 9, 16;\n   3: 0, 0, 0,  0, 25, 72, 49;\n   4: 0, 0, 0,  0,  0,  0, 49, 192, 243, 100;\n   5: 0, 0, 0,  0,  0,  0,  0,   0,  81, 400, 726, 576,  169;\n   6: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0, 121, 720, 1690, 1960, 1125,  256;\n   7: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,  169, 1176, 3375, 5120, 4335,  1944,   361;\n   8: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,  225, 1792, 6069, 11340, 12635,  8400,  3087,   484;\n   9: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,  289,  2592, 10108, 22400, 30870, 27104, 14812,  4608,   625;\n  10: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,    0,     0,   361,  3600, 15876, 40656, 66654, 72576, 52500, 24336, 6561, 784;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-12*n+9+(12*n^2-37*n+28)*x+(9*n^2-30*n+25)*x^2)*P(n,x) = ((4*n^2-4*n+1)*x^2+(16*n^2-17*n+4)*x^3+(21*n^2-25*n+7)*x^4+(9*n^2-12*n+4)*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*P[n - 1, x])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*v[i - 1])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-11b174a1f5e78a68","steps":["(\u03b8+1)\u00b2"],"edges":["e-e0534f334fefeb42"],"text":"Veronese parent of A098172 \u2192 (\u03b8+1)\u00b2"},"headline":"Veronese parent of A098172 \u2192 (\u03b8+1)\u00b2","chain":null}