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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":"0bf90fcf30ca905634728764b3abbb39a27d5929160b8cd52fa9aed020162b96","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-1a86492085c1e21c","role":"sequence","name":"basis_map_A007318 of 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-7813dcd28a577ca9","kind":"fingerprint_related","node_a_id":"n-0ace05ede77c79d0","node_b_id":"n-9a8cb397421dd15e","fingerprint":"f6-4611b6d868b2cab4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4611b6d868b2cab4"},"other_id":"n-0ace05ede77c79d0","other":{"id":"n-0ace05ede77c79d0","name":"theta_plus_one_squared of shift_x_plus_one of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-4611b6d868b2cab4","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","8"]],"card_rows":[["1"],["1"],["3","8"],["3","8"]],"row_sums_preview":["1","1","11","11","93","93","695","695"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-0b9b63070f7f0adc","e-c9162d0502abe241"],"text":"A188440 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A188440 \u2192 basis A007318 \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:     9,     24;\n  3:    27,     72;\n  4:    81,    432,    324;\n  5:   243,   1296,    972;\n  6:   729,   5832,   8748,   3456;\n  7:  2187,  17496,  26244,  10368;\n  8:  6561,  69984, 157464, 124416, 32400;\n  9: 19683, 209952, 472392, 373248, 97200;\n","recurrence_sentence":"The row polynomials satisfy (324+(270*n+189)*x+(126*n^2+54*n+9)*x^2+(18*n^3+33*n^2-9*n-18)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (972+(3402*n-2025)*x+(2538*n^2-486*n-1485)*x^2+(648*n^3+567*n^2-621*n-522)*x^3+(63*n^4+168*n^3+27*n^2-186*n-60)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) - (1944*x+(2754*n+2916)*x^2+(1080*n^2+3726*n+1836)*x^3+(144*n^3+1242*n^2+1854*n+684)*x^4+(6*n^4+132*n^3+426*n^2+468*n+144)*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[((972 + (3402*n - 2025)*x + (2538*n^2 - 486*n - 1485)*x^2 + (648*n^3 + 567*n^2 - 621*n - 522)*x^3 + (63*n^4 + 168*n^3 + 27*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] - (1944*x + (2754*n + 2916)*x^2 + (1080*n^2 + 3726*n + 1836)*x^3 + (144*n^3 + 1242*n^2 + 1854*n + 684)*x^4 + (6*n^4 + 132*n^3 + 426*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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] = ((972 + (3402*n - 2025)*x + (2538*n^2 - 486*n - 1485)*x^2 + (648*n^3 + 567*n^2 - 621*n - 522)*x^3 + (63*n^4 + 168*n^3 + 27*n^2 - 186*n - 60)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] - (1944*x + (2754*n + 2916)*x^2 + (1080*n^2 + 3726*n + 1836)*x^3 + (144*n^3 + 1242*n^2 + 1854*n + 684)*x^4 + (6*n^4 + 132*n^3 + 426*n^2 + 468*n + 144)*x^5 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(324 + (270*n + 189)*x + (126*n^2 + 54*n + 9)*x^2 + (18*n^3 + 33*n^2 - 9*n - 18)*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":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-94bbecf72edd8f15","e-ba987b50aeaf2209"],"text":"A274498 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A274498 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}