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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":"2c6be839139fbbc9852a6dba675334dda238c2b028a44b5145ebda75bc8b82f9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7e99bca717f629d0","role":"sequence","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","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:      0;\n  1:      2;\n  2:     12;\n  3:     54,      32;\n  4:    216,     384;\n  5:    810,    2880,     288;\n  6:   2916,   17280,    5184;\n  7:  10206,   90720,   54432,   2048;\n  8:  34992,  435456,  435456,  49152;\n  9: 118098, 1959552, 2939328, 663552, 12800;\n","recurrence_sentence":"The row polynomials satisfy (243*n-243-(270*n^3-756*n^2+594*n-108)*x+(12*n^5-36*n^4+36*n^3-12*n^2)*x^2)*P(n,x) = (729*n-(810*n^3-1458*n^2-972*n+2592)*x+(36*n^5-72*n^4-612*n^3+1224*n^2+432*n-576)*x^2+(16*n^5-48*n^3-32*n^2)*x^3)*P(n-1,x) + (1458*x-(1620*n^2+324*n)*x^2+(72*n^4+144*n^3+792*n^2+144*n-288)*x^3-(32*n^4+64*n^3+32*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((729*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (36*n^5 - 72*n^4 - 612*n^3 + 1224*n^2 + 432*n - 576)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*P[n - 1, x] + (1458*x - (1620*n^2 + 324*n)*x^2 + (72*n^4 + 144*n^3 + 792*n^2 + 144*n - 288)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*D[P[n - 1, x], x])/(243*n - 243 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (12*n^5 - 36*n^4 + 36*n^3 - 12*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((729*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (36*n^5 - 72*n^4 - 612*n^3 + 1224*n^2 + 432*n - 576)*x^2 + (16*n^5 - 48*n^3 - 32*n^2)*x^3)*v[i - 1] + (1458*x - (1620*n^2 + 324*n)*x^2 + (72*n^4 + 144*n^3 + 792*n^2 + 144*n - 288)*x^3 - (32*n^4 + 64*n^3 + 32*n^2)*x^4)*deriv(v[i - 1], x))/(243*n - 243 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (12*n^5 - 36*n^4 + 36*n^3 - 12*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-01cea33b6485d9cc","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-252e19fd44a7da1c","e-30096dea1fffd844"],"text":"A038220 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A038220 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}