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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":"8468dfedf92554b4af140059d978ff49fdef9b11b6e77a567b6c43d3d3e4b090","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5d4ff1fdd67b7e1d","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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:   4,     4;\n   3:  10,    40,      9;\n   4:  20,   180,    162,      16;\n   5:  35,   560,   1134,     448,      25;\n   6:  56,  1400,   5040,    4480,    1000,      36;\n   7:  84,  3024,  17010,   26880,   13500,    1944,     49;\n   8: 120,  5880,  47628,  117600,  105000,   34020,   3430,     64;\n   9: 165, 10560, 116424,  413952,  577500,  332640,  75460,   5632,   81;\n  10: 220, 17820, 256608, 1241856, 2494800, 2245320, 905520, 152064, 8748, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n-8+(5*n^4-9*n^3-15*n^2+37*n-18)*x+(n^6-3*n^5+10*n^3-15*n^2+9*n-2)*x^2)*P(n,x) = (4*n^2+16*n+16+(5*n^4+6*n^3+21*n^2-52*n-28)*x+(n^6+18*n^4-52*n^3+13*n^2+12*n+20)*x^2+(3*n^6-6*n^5-n^4+12*n^2-8*n)*x^3)*P(n-1,x) + ((8*n+4)*x+(10*n^3-8*n^2-8*n-12)*x^2+(2*n^5-n^4-10*n^3+3*n^2+4*n+8)*x^3-(2*n^5-n^4-5*n^2+4*n)*x^4)*d/dx 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 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*P[n - 1, x] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*n - 2)*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 + 16*n + 16 + (5*n^4 + 6*n^3 + 21*n^2 - 52*n - 28)*x + (n^6 + 18*n^4 - 52*n^3 + 13*n^2 + 12*n + 20)*x^2 + (3*n^6 - 6*n^5 - n^4 + 12*n^2 - 8*n)*x^3)*v[i - 1] + ((8*n + 4)*x + (10*n^3 - 8*n^2 - 8*n - 12)*x^2 + (2*n^5 - n^4 - 10*n^3 + 3*n^2 + 4*n + 8)*x^3 - (2*n^5 - n^4 - 5*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 4*n - 8 + (5*n^4 - 9*n^3 - 15*n^2 + 37*n - 18)*x + (n^6 - 3*n^5 + 10*n^3 - 15*n^2 + 9*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a2304b34e998691c","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-07ea51e83417b459","e-63586532b1ad03dd"],"text":"A062145 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A062145 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}