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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":"26865474e2b2878788423673e0a3328e50fbc1ea9d74907aa6bafb2f171ed555","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b31aa1a3fd93d174","role":"sequence","name":"Delannoy paths counted by number of weak peaks.","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: 1,  16,    12;\n  1: 1,  36,   108,     32;\n  2: 1,  64,   432,    512,      80;\n  3: 1, 100,  1200,   3200,    2000,      192;\n  4: 1, 144,  2700,  12800,   18000,     6912,      448;\n  5: 1, 196,  5292,  39200,   98000,    84672,    21952,      1024;\n  6: 1, 256,  9408, 100352,  392000,   602112,   351232,     65536,     2304;\n  7: 1, 324, 15552, 225792, 1270080,  3048192,  3161088,   1327104,   186624,     5120;\n  8: 1, 400, 24300, 460800, 3528000, 12192768, 19756800,  14745600,  4665600,   512000,   11264;\n  9: 1, 484, 36300, 871200, 8712000, 40981248, 95622912, 111513600, 62726400, 15488000, 1362944, 24576;\n","recurrence_sentence":"The row polynomials satisfy -(n+2-(2*n^3+12*n^2+24*n+16)*x)*P(n,x) = -(n+2-(2*n^3+12*n^2+12*n)*x-(12*n^3+80*n^2+164*n+104)*x^2)*P(n-1,x) - (2*x-(4*n^2+22*n+32)*x^2+(8*n^2+44*n+56)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 12*x^2.","mathematica":"P[0, x_] := 1 + 16*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 2 - (2*n^3 + 12*n^2 + 12*n)*x - (12*n^3 + 80*n^2 + 164*n + 104)*x^2)*P[n - 1, x] - (2*x - (4*n^2 + 22*n + 32)*x^2 + (8*n^2 + 44*n + 56)*x^3)*D[P[n - 1, x], x])/(-n - 2 + (2*n^3 + 12*n^2 + 24*n + 16)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n + 2 - (2*n^3 + 12*n^2 + 12*n)*x - (12*n^3 + 80*n^2 + 164*n + 104)*x^2)*v[i - 1] - (2*x - (4*n^2 + 22*n + 32)*x^2 + (8*n^2 + 44*n + 56)*x^3)*deriv(v[i - 1], x))/(-n - 2 + (2*n^3 + 12*n^2 + 24*n + 16)*x)); 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-b31aa1a3fd93d174","steps":["\u03b8+1"],"edges":["e-252d74b751e01a16"],"text":"A133214 \u2192 \u03b8+1"},"headline":"A133214 \u2192 \u03b8+1","chain":null}