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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":"edca54ecffa34d48569df4693a0153b719f2ae33742e1b13cb80a792aefde47c","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-4246e71c9dd6d2b5","role":"sequence","name":"basis_map_A013610 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,    12;\n  3:    7,    48,     54;\n  4:   17,   168,    378,     216;\n  5:   41,   528,   1782,    2160,     810;\n  6:   99,  1572,   7074,   13176,   10530,    2916;\n  7:  239,  4512,  25380,   63936,   79380,   46656,   10206;\n  8:  577, 12624,  85212,  270864,  458460,  419904,  193914,  34992;\n  9: 1393, 34656, 272916, 1048032, 2251800, 2822688, 2030994, 769824, 118098;\n","recurrence_sentence":"The row polynomials satisfy -(96-(72*n-648)*x-(216*n^2-108*n-1242)*x^2-(162*n^3+162*n^2-810*n-324)*x^3-(243*n^2-729*n+486)*x^4)*P(n,x) = -(192*n-96+(432*n^2+792*n-648)*x+(216*n^3+1512*n^2+648*n-1242)*x^2+(486*n^3+1134*n^2-486*n-324)*x^3+(243*n^2-729*n+486)*x^4)*P(n-1,x) + (64+(192*n+672)*x+(144*n^2+1584*n+2664)*x^2+(864*n^2+4752*n+4860)*x^3+(1458*n^2+6318*n+3888)*x^4+(486*n^2+3645*n+972)*x^5+729*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[(-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*P[n - 1, x] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*D[P[n - 1, x], x])/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*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] = (-(192*n - 96 + (432*n^2 + 792*n - 648)*x + (216*n^3 + 1512*n^2 + 648*n - 1242)*x^2 + (486*n^3 + 1134*n^2 - 486*n - 324)*x^3 + (243*n^2 - 729*n + 486)*x^4)*v[i - 1] + (64 + (192*n + 672)*x + (144*n^2 + 1584*n + 2664)*x^2 + (864*n^2 + 4752*n + 4860)*x^3 + (1458*n^2 + 6318*n + 3888)*x^4 + (486*n^2 + 3645*n + 972)*x^5 + 729*n*x^6)*deriv(v[i - 1], x))/(-96 + (72*n - 648)*x + (216*n^2 - 108*n - 1242)*x^2 + (162*n^3 + 162*n^2 - 810*n - 324)*x^3 + (243*n^2 - 729*n + 486)*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-a1540a6d32c29f8f","steps":["basis A013610","\u03b8+1"],"edges":["e-251e56a692041c9f","e-2d5962bc9e95ac1d"],"text":"A266213 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A266213 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}