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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":"8ea6b7345ae7d13e5fac6bcff0c7484caef0c3042fa89838a3127e007c17acc0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-726f071d833f296a","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-53758eb0aca22f39","kind":"fingerprint_related","node_a_id":"n-3a17caa08883823f","node_b_id":"n-48b8e805a964eabc","fingerprint":"f6-b7c8d43fd9e53efa","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7c8d43fd9e53efa"},"other_id":"n-3a17caa08883823f","other":{"id":"n-3a17caa08883823f","name":"theta_plus_one_squared of basis_map_A007318 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-b7c8d43fd9e53efa","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","8"],["6","40","36"]],"card_rows":[["1"],["2","8"],["6","40","36"],["16","160","288","128"]],"row_sums_preview":["1","10","82","592","3976","25408","156608","939008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-23c3ede392db8800","e-bc078aff5db44b1f"],"text":"A099040 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A099040 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    2,      8;\n   3:    6,     40,      36;\n   4:   16,    160,     288,     128;\n   5:   44,    576,    1548,    1408,     400;\n   6:  120,   1952,    6984,    9600,    5600,    1152;\n   7:  328,   6368,   28440,   52352,   46400,   19584,    3136;\n   8:  896,  20224,  108288,  249856,  297600,  191232,   62720,    8192;\n   9: 2448,  62976,  393120, 1090560, 1632400, 1412352,  705600,  188416,  20736;\n  10: 6688, 193152, 1377216, 4463616, 8048800, 8765568, 5901952, 2400256, 539136, 51200;\n","recurrence_sentence":"The row polynomials satisfy (36+(108*n-120)*x+(78*n^2-126*n+16)*x^2+(18*n^3-22*n^2-60*n+64)*x^3+(n^4+2*n^3-29*n^2+26*n+8)*x^4-(2*n^3-6*n+4)*x^5)*P(n,x) = (72+(216*n+48)*x+(156*n^2+276*n-88)*x^2+(36*n^3+220*n^2-40*n-64)*x^3+(2*n^4+48*n^3+30*n^2-144*n+16)*x^4+(2*n^4+8*n^3-46*n^2-44*n+16)*x^5-(4*n^3+12*n^2)*x^6)*P(n-2,x) + (72+(216*n+48)*x+(156*n^2+192*n-88)*x^2+(36*n^3+184*n^2-176*n-64)*x^3+(2*n^4+44*n^3-18*n^2-188*n+16)*x^4+(2*n^4+4*n^3-54*n^2-36*n+16)*x^5-(4*n^3+8*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((72 + (216*n + 48)*x + (156*n^2 + 276*n - 88)*x^2 + (36*n^3 + 220*n^2 - 40*n - 64)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (2*n^4 + 8*n^3 - 46*n^2 - 44*n + 16)*x^5 - (4*n^3 + 12*n^2)*x^6)*P[n - 2, x] + (72 + (216*n + 48)*x + (156*n^2 + 192*n - 88)*x^2 + (36*n^3 + 184*n^2 - 176*n - 64)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (2*n^4 + 4*n^3 - 54*n^2 - 36*n + 16)*x^5 - (4*n^3 + 8*n^2)*x^6)*P[n - 1, x])/(36 + (108*n - 120)*x + (78*n^2 - 126*n + 16)*x^2 + (18*n^3 - 22*n^2 - 60*n + 64)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (2*n^3 - 6*n + 4)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 8*x; for(n = 3, N, my(i = n); v[i] = ((72 + (216*n + 48)*x + (156*n^2 + 276*n - 88)*x^2 + (36*n^3 + 220*n^2 - 40*n - 64)*x^3 + (2*n^4 + 48*n^3 + 30*n^2 - 144*n + 16)*x^4 + (2*n^4 + 8*n^3 - 46*n^2 - 44*n + 16)*x^5 - (4*n^3 + 12*n^2)*x^6)*v[i - 2] + (72 + (216*n + 48)*x + (156*n^2 + 192*n - 88)*x^2 + (36*n^3 + 184*n^2 - 176*n - 64)*x^3 + (2*n^4 + 44*n^3 - 18*n^2 - 188*n + 16)*x^4 + (2*n^4 + 4*n^3 - 54*n^2 - 36*n + 16)*x^5 - (4*n^3 + 8*n^2)*x^6)*v[i - 1])/(36 + (108*n - 120)*x + (78*n^2 - 126*n + 16)*x^2 + (18*n^3 - 22*n^2 - 60*n + 64)*x^3 + (n^4 + 2*n^3 - 29*n^2 + 26*n + 8)*x^4 - (2*n^3 - 6*n + 4)*x^5)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-081205c1b1ff787d","e-5b297c40b08a8f01"],"text":"A063967 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A063967 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}