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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":"b1c292d74217ffd036bfa6e9a8852110baff6da036e8a9739247f519ca243345","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","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:      1;\n   2:      3,       4;\n   3:     11,      32,       12;\n   4:     45,     204,      180,       32;\n   5:    197,    1216,     1752,      768,       80;\n   6:    903,    7060,    14280,    10560,     2800,      192;\n   7:   4279,   40512,   106020,   112640,    51600,     9216,      448;\n   8:  20793,  231196,   744156,  1040480,   697200,   219072,    28224,    1024;\n   9: 103049, 1316096,  5034288,  8770048,  7841120,  3655680,   840448,   81920,   2304;\n  10: 518859, 7484580, 33192720, 69417600, 78170400, 49182336, 16972032, 2985984, 228096, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(2*n+2-(4*n^2+4*n)*x-(4*n^3-4*n)*x^2)*P(n,x) = -(10*n-2-(16*n^2-16*n+16)*x-(20*n^3+24*n^2-44*n)*x^2-(24*n^3-16*n^2-24*n+16)*x^3)*P(n-1,x) - (1-4*n*x-(4*n^2+2*n+2)*x^2+16*n*x^3+(16*n^2+8*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(10*n - 2 - (16*n^2 - 16*n + 16)*x - (20*n^3 + 24*n^2 - 44*n)*x^2 - (24*n^3 - 16*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - (1 - 4*n*x - (4*n^2 + 2*n + 2)*x^2 + 16*n*x^3 + (16*n^2 + 8*n - 8)*x^4)*D[P[n - 1, x], x])/(-2*n - 2 + (4*n^2 + 4*n)*x + (4*n^3 - 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(10*n - 2 - (16*n^2 - 16*n + 16)*x - (20*n^3 + 24*n^2 - 44*n)*x^2 - (24*n^3 - 16*n^2 - 24*n + 16)*x^3)*v[i - 1] - (1 - 4*n*x - (4*n^2 + 2*n + 2)*x^2 + 16*n*x^3 + (16*n^2 + 8*n - 8)*x^4)*deriv(v[i - 1], x))/(-2*n - 2 + (4*n^2 + 4*n)*x + (4*n^3 - 4*n)*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-0d75f6272d2a2a84","steps":["basis A013609","\u03b8+1"],"edges":["e-077b645e97fdde1b","e-ec8d721b8779aeb7"],"text":"A114656 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A114656 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}