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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":"11f9ab153134831bf8314bc530e12a2194b29c65e1abec501681f9ed1ebaa741","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-7c641d81e21514ce","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","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":[{"id":"l-e418885e33130eb7","kind":"fingerprint_related","node_a_id":"n-72ed2075ed1a9d58","node_b_id":"n-33cdc2fbd5134453","fingerprint":"f6-5367d60e6d4311ec","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5367d60e6d4311ec"},"other_id":"n-72ed2075ed1a9d58","other":{"id":"n-72ed2075ed1a9d58","name":"theta_plus_one_squared of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-5367d60e6d4311ec","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","8"]],"row_sums_preview":["1","1","9","9","53","53","261","261"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["(\u03b8+1)\u00b2"],"edges":["e-0d38730cf56d64c0"],"text":"A188440 \u2192 (\u03b8+1)\u00b2"},"headline":"A188440 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:   1;\n  1:   3;\n  2:   3,   24;\n  3:   9,   72;\n  4:   9,  144,   324;\n  5:  27,  432,   972;\n  6:  27,  648,  2916,   3456;\n  7:  81, 1944,  8748,  10368;\n  8:  81, 2592, 17496,  41472, 32400;\n  9: 243, 7776, 52488, 124416, 97200;\n","recurrence_sentence":"The row polynomials satisfy (4+(10*n+7)*x+(14*n^2+6*n+1)*x^2+(6*n^3+11*n^2-3*n-6)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = -(8*n-20+(34*n^2-110*n+25)*x+(40*n^3-172*n^2+14*n+55)*x^2+(16*n^4-90*n^3-97*n^2+73*n+74)*x^3+(2*n^5-21*n^4-62*n^3-13*n^2+62*n+20)*x^4-(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) + (2+(12*n-6)*x+(22*n^2-12*n-42)*x^2+(12*n^3+14*n^2-144*n-76)*x^3+(2*n^4+12*n^3-116*n^2-210*n-84)*x^4+(2*n^4-36*n^3-138*n^2-156*n-48)*x^5-(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 20 + (34*n^2 - 110*n + 25)*x + (40*n^3 - 172*n^2 + 14*n + 55)*x^2 + (16*n^4 - 90*n^3 - 97*n^2 + 73*n + 74)*x^3 + (2*n^5 - 21*n^4 - 62*n^3 - 13*n^2 + 62*n + 20)*x^4 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] + (2 + (12*n - 6)*x + (22*n^2 - 12*n - 42)*x^2 + (12*n^3 + 14*n^2 - 144*n - 76)*x^3 + (2*n^4 + 12*n^3 - 116*n^2 - 210*n - 84)*x^4 + (2*n^4 - 36*n^3 - 138*n^2 - 156*n - 48)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n - 20 + (34*n^2 - 110*n + 25)*x + (40*n^3 - 172*n^2 + 14*n + 55)*x^2 + (16*n^4 - 90*n^3 - 97*n^2 + 73*n + 74)*x^3 + (2*n^5 - 21*n^4 - 62*n^3 - 13*n^2 + 62*n + 20)*x^4 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] + (2 + (12*n - 6)*x + (22*n^2 - 12*n - 42)*x^2 + (12*n^3 + 14*n^2 - 144*n - 76)*x^3 + (2*n^4 + 12*n^3 - 116*n^2 - 210*n - 84)*x^4 + (2*n^4 - 36*n^3 - 138*n^2 - 156*n - 48)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(4 + (10*n + 7)*x + (14*n^2 + 6*n + 1)*x^2 + (6*n^3 + 11*n^2 - 3*n - 6)*x^3 + (n^4 + 2*n^3 + n^2)*x^4)); 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-7c641d81e21514ce","steps":["(\u03b8+1)\u00b2"],"edges":["e-be9753f82aa1d431"],"text":"A274498 \u2192 (\u03b8+1)\u00b2"},"headline":"A274498 \u2192 (\u03b8+1)\u00b2","chain":null}