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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":"a51ce5b6637d6be73b361861b018f851ba91f97d68bc2736040bcdb0b8707aaa","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-117b55f7953e214b","role":"sequence","name":"basis_map_A013610 of Square the entries of Pascal's triangle.","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,       6;\n  2:     6,      36,       27;\n  3:    20,     180,      324,      108;\n  4:    70,     840,     2430,     2160,       405;\n  5:   252,    3780,    15120,    22680,     12150,      1458;\n  6:   924,   16632,    85050,   181440,    170100,     61236,     5103;\n  7:  3432,   72072,   449064,  1247400,   1701000,   1102248,   285768,    17496;\n  8: 12870,  308880,  2270268,  7783776,  14033250,  13471920,  6429780,  1259712,   59049;\n  9: 48620, 1312740, 11119680, 45405360, 102162060, 131351220, 94303440, 34642080, 5314410, 196830;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3*n^2*x)*P(n,x) = (8*n-4+(12*n^2+30*n-24)*x+(27*n^2+18*n-27)*x^2)*P(n-1,x) - (4*x+(6*n+21)*x^2+(18*n+27)*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 - 4 + (12*n^2 + 30*n - 24)*x + (27*n^2 + 18*n - 27)*x^2)*P[n - 1, x] - (4*x + (6*n + 21)*x^2 + (18*n + 27)*x^3)*D[P[n - 1, x], x])/(2*n + 3*n^2*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 - 4 + (12*n^2 + 30*n - 24)*x + (27*n^2 + 18*n - 27)*x^2)*v[i - 1] - (4*x + (6*n + 21)*x^2 + (18*n + 27)*x^3)*deriv(v[i - 1], x))/(2*n + 3*n^2*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-b13531ad79e7c629","steps":["basis A013610","\u03b8+1"],"edges":["e-0e684cdba462224e","e-dc07883d56c1dcd5"],"text":"A008459 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A008459 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}