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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":"6df64470665c7adc749ed095fcc729e7bd3a1973d66fb7fc7753b44751642047","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-3594f9d1943ddb79","role":"sequence","name":"basis_map_A013610 of delannoy","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-516fd5ababb6e59c","kind":"fingerprint_related","node_a_id":"n-1ef27b3780d3024e","node_b_id":"n-967c8d9a24748c8e","fingerprint":"f6-8855fe109ba04775","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8855fe109ba04775"},"other_id":"n-967c8d9a24748c8e","other":{"id":"n-967c8d9a24748c8e","name":"theta_plus_one of basis_map_A013610 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","first_index":1,"fingerprint":"f6-8855fe109ba04775","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","6"],["5","30","27"]],"card_rows":[["1"],["2","6"],["5","30","27"],["12","108","216","108"]],"row_sums_preview":["1","8","62","444","3050","20344","132838","853428"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013610","\u03b8+1"],"edges":["e-79afa45c031edfc6","e-7bd487fecae15b8d"],"text":"A008288 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A008288 \u2192 basis A013610 \u2192 \u03b8+1","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:    2,     6;\n  2:    5,    30,     27;\n  3:   12,   108,    216,     108;\n  4:   29,   348,   1080,    1188,     405;\n  5:   70,  1050,   4428,    7668,    5670,    1458;\n  6:  169,  3042,  16227,   38556,   44955,   24786,    5103;\n  7:  408,  8568,  55296,  167400,  268920,  233280,  102060,   17496;\n  8:  985, 23640, 179064,  659448, 1355130, 1621296, 1112454,  402408,   59049;\n  9: 2378, 64206, 558360, 2423736, 6079860, 9322452, 8817984, 4986360, 1535274, 196830;\n","recurrence_sentence":"The row polynomials satisfy (108*n*x+(162*n^2+324*n)*x^2+(81*n^3+243*n^2+162*n)*x^3)*P(n,x) = -(96*n-96+(216*n^2+324*n-756)*x+(108*n^3+702*n^2-378*n-2052)*x^2+(243*n^3+162*n^2-2025*n-2268)*x^3-(486*n^2+1458*n+972)*x^4)*P(n-1,x) + (32+(96*n+336)*x+(72*n^2+792*n+1332)*x^2+(432*n^2+2268*n+2430)*x^3+(729*n^2+2511*n+1944)*x^4+(243*n^2+729*n+486)*x^5)*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[(-(96*n - 96 + (216*n^2 + 324*n - 756)*x + (108*n^3 + 702*n^2 - 378*n - 2052)*x^2 + (243*n^3 + 162*n^2 - 2025*n - 2268)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*P[n - 1, x] + (32 + (96*n + 336)*x + (72*n^2 + 792*n + 1332)*x^2 + (432*n^2 + 2268*n + 2430)*x^3 + (729*n^2 + 2511*n + 1944)*x^4 + (243*n^2 + 729*n + 486)*x^5)*D[P[n - 1, x], x])/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)]];\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] = (-(96*n - 96 + (216*n^2 + 324*n - 756)*x + (108*n^3 + 702*n^2 - 378*n - 2052)*x^2 + (243*n^3 + 162*n^2 - 2025*n - 2268)*x^3 - (486*n^2 + 1458*n + 972)*x^4)*v[i - 1] + (32 + (96*n + 336)*x + (72*n^2 + 792*n + 1332)*x^2 + (432*n^2 + 2268*n + 2430)*x^3 + (729*n^2 + 2511*n + 1944)*x^4 + (243*n^2 + 729*n + 486)*x^5)*deriv(v[i - 1], x))/(108*n*x + (162*n^2 + 324*n)*x^2 + (81*n^3 + 243*n^2 + 162*n)*x^3)); 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-e00a1cb428776eaa","steps":["basis A013610","\u03b8+1"],"edges":["e-470609786340cbfa","e-a7b356f9dcf5bfbc"],"text":"delannoy \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"delannoy \u2192 basis A013610 \u2192 \u03b8+1","chain":null}