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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":"41ee7891f185347e7b4d4e42fa80e2f9abcad998a31a383db6601d661d82d582","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-45d316ccbe3d57d1","role":"sequence","name":"basis_map_A013609 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,     8;\n  3:    7,    32,     24;\n  4:   17,   112,    168,     64;\n  5:   41,   352,    792,    640,    160;\n  6:   99,  1048,   3144,   3904,   2080,    384;\n  7:  239,  3008,  11280,  18944,  15680,   6144,    896;\n  8:  577,  8416,  37872,  80256,  90560,  55296,  17024,  2048;\n  9: 1393, 23104, 121296, 310528, 444800, 371712, 178304, 45056, 4608;\n","recurrence_sentence":"The row polynomials satisfy -(4-(2*n-18)*x-(4*n^2-2*n-23)*x^2-(2*n^3+2*n^2-10*n-4)*x^3-(2*n^2-6*n+4)*x^4)*P(n,x) = -(8*n-4+(12*n^2+22*n-18)*x+(4*n^3+28*n^2+12*n-23)*x^2+(6*n^3+14*n^2-6*n-4)*x^3+(2*n^2-6*n+4)*x^4)*P(n-1,x) + (4+(8*n+28)*x+(4*n^2+44*n+74)*x^2+(16*n^2+88*n+90)*x^3+(18*n^2+78*n+48)*x^4+(4*n^2+30*n+8)*x^5+4*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[(-(8*n - 4 + (12*n^2 + 22*n - 18)*x + (4*n^3 + 28*n^2 + 12*n - 23)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)*P[n - 1, x] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 74)*x^2 + (16*n^2 + 88*n + 90)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (4*n^2 + 30*n + 8)*x^5 + 4*n*x^6)*D[P[n - 1, x], x])/(-4 + (2*n - 18)*x + (4*n^2 - 2*n - 23)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (2*n^2 - 6*n + 4)*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] = (-(8*n - 4 + (12*n^2 + 22*n - 18)*x + (4*n^3 + 28*n^2 + 12*n - 23)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (2*n^2 - 6*n + 4)*x^4)*v[i - 1] + (4 + (8*n + 28)*x + (4*n^2 + 44*n + 74)*x^2 + (16*n^2 + 88*n + 90)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (4*n^2 + 30*n + 8)*x^5 + 4*n*x^6)*deriv(v[i - 1], x))/(-4 + (2*n - 18)*x + (4*n^2 - 2*n - 23)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (2*n^2 - 6*n + 4)*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 A013609","\u03b8+1"],"edges":["e-064ccd10ffc87509","e-89cab765d6e5d09c"],"text":"A035607 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A035607 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}