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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":"49362fd43772bed1a4f1d14b1bcb937857145472a4a077141bc1f7d15edeea4b","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-f91986a1e2f81fd1","role":"sequence","name":"basis_map_A013611 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","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:    1;\n  2:    3,    16;\n  3:    7,    64,     96;\n  4:   17,   224,    672,     512;\n  5:   41,   704,   3168,    5120,    2560;\n  6:   99,  2096,  12576,   31232,   33280,    12288;\n  7:  239,  6016,  45120,  151552,  250880,   196608,    57344;\n  8:  577, 16832, 151488,  642048, 1448960,  1769472,  1089536,  262144;\n  9: 1393, 46208, 485184, 2484224, 7116800, 11894784, 11411456, 5767168, 1179648;\n","recurrence_sentence":"The row polynomials satisfy -(2-(2*n-18)*x-(8*n^2-4*n-46)*x^2-(8*n^3+8*n^2-40*n-16)*x^3-(16*n^2-48*n+32)*x^4)*P(n,x) = -(4*n-2+(12*n^2+22*n-18)*x+(8*n^3+56*n^2+24*n-46)*x^2+(24*n^3+56*n^2-24*n-16)*x^3+(16*n^2-48*n+32)*x^4)*P(n-1,x) + (1+(4*n+14)*x+(4*n^2+44*n+74)*x^2+(32*n^2+176*n+180)*x^3+(72*n^2+312*n+192)*x^4+(32*n^2+240*n+64)*x^5+64*n*x^6)*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[(-(4*n - 2 + (12*n^2 + 22*n - 18)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (24*n^3 + 56*n^2 - 24*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)*P[n - 1, x] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (72*n^2 + 312*n + 192)*x^4 + (32*n^2 + 240*n + 64)*x^5 + 64*n*x^6)*D[P[n - 1, x], x])/(-2 + (2*n - 18)*x + (8*n^2 - 4*n - 46)*x^2 + (8*n^3 + 8*n^2 - 40*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)]];\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] = (-(4*n - 2 + (12*n^2 + 22*n - 18)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (24*n^3 + 56*n^2 - 24*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)*v[i - 1] + (1 + (4*n + 14)*x + (4*n^2 + 44*n + 74)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (72*n^2 + 312*n + 192)*x^4 + (32*n^2 + 240*n + 64)*x^5 + 64*n*x^6)*deriv(v[i - 1], x))/(-2 + (2*n - 18)*x + (8*n^2 - 4*n - 46)*x^2 + (8*n^3 + 8*n^2 - 40*n - 16)*x^3 + (16*n^2 - 48*n + 32)*x^4)); 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-e073e5e0b021f36a","steps":["basis A013611","\u03b8+1"],"edges":["e-8337e61ed504a271","e-cf9ca9a35cd26222"],"text":"A035607 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A035607 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}