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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":"b27f0991dc2892a2fec8a2dd23aabe13c0d0566b2be114f104d7623f0790ded8","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-a8bf9bc375c66ce8","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: 0, 0, 9;\n   2: 0, 0, 0, 0, 25, 36;\n   3: 0, 0, 0, 0,  0,  0, 49, 128, 81;\n   4: 0, 0, 0, 0,  0,  0,  0,   0, 81, 300, 363, 144;\n   5: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0, 121, 576, 1014, 784,  225;\n   6: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,  169, 980, 2250, 2560, 1445,  324;\n   7: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,  225, 1536, 4335, 6480, 5415,  2400,   441;\n   8: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,  289, 2268, 7581, 14000, 15435, 10164,  3703,   576;\n   9: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,  361,  3200, 12348, 27104, 37030, 32256, 17500,  5408,   729;\n  10: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,    0,     0,   441,  4356, 19044, 48384, 78750, 85176, 61236, 28224, 7569, 900;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-4*n+1+(12*n^2-17*n+4)*x+(9*n^2-18*n+9)*x^2)*P(n,x) = ((4*n^2+4*n+1)*x^2+(16*n^2+11*n)*x^3+(21*n^2+7*n-1)*x^4+9*n^2*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 9*x^2.","mathematica":"P[1, x_] := 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*P[n - 1, x])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 9*x^2; for(n = 2, N, my(i = n); v[i] = (((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*v[i - 1])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*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-a8bf9bc375c66ce8","steps":["(\u03b8+1)\u00b2"],"edges":["e-f61ac1f99d3946a9"],"text":"Veronese parent of A098172 \u2192 (\u03b8+1)\u00b2"},"headline":"Veronese parent of A098172 \u2192 (\u03b8+1)\u00b2","chain":null}