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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":"7399bec93a8d1b4236b7a12364d255d1b9f92e92690833b31e73d2a2c1dad571","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-9ac33c0e4aa39b7d","role":"sequence","name":"basis_map_A007318 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,     4;\n  3:    7,    16,     6;\n  4:   17,    56,    42,     8;\n  5:   41,   176,   198,    80,    10;\n  6:   99,   524,   786,   488,   130,    12;\n  7:  239,  1504,  2820,  2368,   980,   192,   14;\n  8:  577,  4208,  9468, 10032,  5660,  1728,  266,  16;\n  9: 1393, 11552, 30324, 38816, 27800, 11616, 2786, 352, 18;\n","recurrence_sentence":"The row polynomials satisfy -(32-(8*n-72)*x-(8*n^2-4*n-46)*x^2-(2*n^3+2*n^2-10*n-4)*x^3-(n^2-3*n+2)*x^4)*P(n,x) = -(64*n-32+(48*n^2+88*n-72)*x+(8*n^3+56*n^2+24*n-46)*x^2+(6*n^3+14*n^2-6*n-4)*x^3+(n^2-3*n+2)*x^4)*P(n-1,x) + (64+(64*n+224)*x+(16*n^2+176*n+296)*x^2+(32*n^2+176*n+180)*x^3+(18*n^2+78*n+48)*x^4+(2*n^2+15*n+4)*x^5+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[(-(64*n - 32 + (48*n^2 + 88*n - 72)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (n^2 - 3*n + 2)*x^4)*P[n - 1, x] + (64 + (64*n + 224)*x + (16*n^2 + 176*n + 296)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (2*n^2 + 15*n + 4)*x^5 + n*x^6)*D[P[n - 1, x], x])/(-32 + (8*n - 72)*x + (8*n^2 - 4*n - 46)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (n^2 - 3*n + 2)*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] = (-(64*n - 32 + (48*n^2 + 88*n - 72)*x + (8*n^3 + 56*n^2 + 24*n - 46)*x^2 + (6*n^3 + 14*n^2 - 6*n - 4)*x^3 + (n^2 - 3*n + 2)*x^4)*v[i - 1] + (64 + (64*n + 224)*x + (16*n^2 + 176*n + 296)*x^2 + (32*n^2 + 176*n + 180)*x^3 + (18*n^2 + 78*n + 48)*x^4 + (2*n^2 + 15*n + 4)*x^5 + n*x^6)*deriv(v[i - 1], x))/(-32 + (8*n - 72)*x + (8*n^2 - 4*n - 46)*x^2 + (2*n^3 + 2*n^2 - 10*n - 4)*x^3 + (n^2 - 3*n + 2)*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 A007318","\u03b8+1"],"edges":["e-017120e86183432c","e-5c0e106a96551815"],"text":"A035607 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A035607 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}