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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":"9a87a8b2de27bf1887961811c31520f47d459bf74580b6d77af9b77994dbb167","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-1b336857eed4ab2c","role":"sequence","name":"derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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;\n   2:   1;\n   3:   4,    8;\n   4:  10,   48,    27;\n   5:  20,  168,   216,     64;\n   6:  35,  448,   972,    640,    125;\n   7:  56, 1008,  3240,   3520,   1500,    216;\n   8:  84, 2016,  8910,  14080,   9750,   3024,   343;\n   9: 120, 3696, 21384,  45760,  45500,  22680,  5488,  512;\n  10: 165, 6336, 46332, 128128, 170625, 120960, 46648, 9216, 729;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16+(4*n^3-16*n^2+8*n+16)*x+(2*n^5-12*n^4+24*n^3-16*n^2)*x^2)*P(n,x) = (8*n+8+(4*n^3-4*n^2+8)*x+(2*n^5-6*n^4+2*n^3+8*n^2-6*n-4)*x^2+(n^5-n^4-n^3+n^2)*x^3)*P(n-1,x) + (16*x+(8*n^2-8*n-4)*x^2+(4*n^4-8*n^3+6*n^2-2*n-2)*x^3+(n^4-2*n^3+n^2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 + (4*n^3 - 4*n^2 + 8)*x + (2*n^5 - 6*n^4 + 2*n^3 + 8*n^2 - 6*n - 4)*x^2 + (n^5 - n^4 - n^3 + n^2)*x^3)*P[n - 1, x] + (16*x + (8*n^2 - 8*n - 4)*x^2 + (4*n^4 - 8*n^3 + 6*n^2 - 2*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 1, x], x])/(8*n - 16 + (4*n^3 - 16*n^2 + 8*n + 16)*x + (2*n^5 - 12*n^4 + 24*n^3 - 16*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n + 8 + (4*n^3 - 4*n^2 + 8)*x + (2*n^5 - 6*n^4 + 2*n^3 + 8*n^2 - 6*n - 4)*x^2 + (n^5 - n^4 - n^3 + n^2)*x^3)*v[i - 1] + (16*x + (8*n^2 - 8*n - 4)*x^2 + (4*n^4 - 8*n^3 + 6*n^2 - 2*n - 2)*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(v[i - 1], x))/(8*n - 16 + (4*n^3 - 16*n^2 + 8*n + 16)*x + (2*n^5 - 12*n^4 + 24*n^3 - 16*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-976cdb6e4a693ec8","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-1dbefba2139b076a","e-2e607f4ed4a7abdb"],"text":"A078812 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A078812 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}