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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"9390701476d364410251149d9a0fe84713e9875cd605efd14fadcb26ff24cfbb","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-8b333ce67e8457dc","role":"sequence","name":"basis_map_A013611 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 double rises (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:      3,       16;\n  2:     11,      112,       192;\n  3:     45,      672,      2304,       2048;\n  4:    197,     3872,     19968,      36864,      20480;\n  5:    903,    22000,    151680,     430080,     512000,      196608;\n  6:   4279,   124368,   1074240,    4126720,    7680000,     6488064,     1835008;\n  7:  20793,   702016,   7289856,   35409920,   90316800,   121110528,    77070336,    16777216;\n  8: 103049,  3962944,  48082944,  282705920,  919224320,  1706557440,  1746927616,   872415232,  150994944;\n  9: 518859, 22388688, 310839552, 2146615296, 8500101120, 20263993344, 28978446336, 23555211264, 9512681472, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy (4*n+8+(32*n^2+96*n+64)*x+(64*n^3+192*n^2+128*n)*x^2)*P(n,x) = (28*n+8+(256*n^2+448*n+64)*x+(448*n^3+2112*n^2+1664*n)*x^2+(1536*n^3+3584*n^2+1024*n)*x^3)*P(n-1,x) - (1+(16*n+28)*x+(32*n^2+272*n+256)*x^2+(384*n^2+1472*n+896)*x^3+(1024*n^2+2560*n+1024)*x^4)*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[((28*n + 8 + (256*n^2 + 448*n + 64)*x + (448*n^3 + 2112*n^2 + 1664*n)*x^2 + (1536*n^3 + 3584*n^2 + 1024*n)*x^3)*P[n - 1, x] - (1 + (16*n + 28)*x + (32*n^2 + 272*n + 256)*x^2 + (384*n^2 + 1472*n + 896)*x^3 + (1024*n^2 + 2560*n + 1024)*x^4)*D[P[n - 1, x], x])/(4*n + 8 + (32*n^2 + 96*n + 64)*x + (64*n^3 + 192*n^2 + 128*n)*x^2)]];\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] = ((28*n + 8 + (256*n^2 + 448*n + 64)*x + (448*n^3 + 2112*n^2 + 1664*n)*x^2 + (1536*n^3 + 3584*n^2 + 1024*n)*x^3)*v[i - 1] - (1 + (16*n + 28)*x + (32*n^2 + 272*n + 256)*x^2 + (384*n^2 + 1472*n + 896)*x^3 + (1024*n^2 + 2560*n + 1024)*x^4)*deriv(v[i - 1], x))/(4*n + 8 + (32*n^2 + 96*n + 64)*x + (64*n^3 + 192*n^2 + 128*n)*x^2)); 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-8ce61d302cec295b","steps":["basis A013611","\u03b8+1"],"edges":["e-034e0d2a0f7f14a7","e-4806be79538dd3ab"],"text":"A114687 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A114687 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}