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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":"05a79b59cf11d38e7e9d23120eb530702c634898e84b284b79d0c081456a1006","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-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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;\n  1:     2,      2;\n  2:     5,     10,      3;\n  3:    14,     42,     27,      4;\n  4:    42,    168,    168,     56,      5;\n  5:   132,    660,    900,    480,    100,      6;\n  6:   429,   2574,   4455,   3300,   1125,    162,     7;\n  7:  1430,  10010,  21021,  20020,   9625,   2310,   245,    8;\n  8:  4862,  38896,  96096, 112112,  70070,  24024,  4312,  352,   9;\n  9: 16796, 151164, 429624, 594048, 458640, 206388, 53508, 7488, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+6*n+4+(n^3+3*n^2+2*n)*x)*P(n,x) = (8*n^2+12*n+4+(4*n^3+18*n^2+14*n)*x+(3*n^3+7*n^2+2*n)*x^2)*P(n-1,x) - ((4*n+4)*x+(2*n^2+9*n+6)*x^2+(2*n^2+5*n+2)*x^3)*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^2 + 12*n + 4 + (4*n^3 + 18*n^2 + 14*n)*x + (3*n^3 + 7*n^2 + 2*n)*x^2)*P[n - 1, x] - ((4*n + 4)*x + (2*n^2 + 9*n + 6)*x^2 + (2*n^2 + 5*n + 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 + 6*n + 4 + (n^3 + 3*n^2 + 2*n)*x)]];\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^2 + 12*n + 4 + (4*n^3 + 18*n^2 + 14*n)*x + (3*n^3 + 7*n^2 + 2*n)*x^2)*v[i - 1] - ((4*n + 4)*x + (2*n^2 + 9*n + 6)*x^2 + (2*n^2 + 5*n + 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 + 6*n + 4 + (n^3 + 3*n^2 + 2*n)*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-57b4b14604f4c0d7","steps":["\u03b8+1"],"edges":["e-50516f7b15ad2326"],"text":"A126216 \u2192 \u03b8+1"},"headline":"A126216 \u2192 \u03b8+1","chain":null}