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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":"4ae65a3327e101158328a492d04af3fed642e4c84d54714e1538594bd3ea4db7","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-b6aca576a689e9f0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","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":[{"id":"l-336c1ee014ca6e65","kind":"fingerprint_related","node_a_id":"n-05fe2267b7e2ee62","node_b_id":"n-44e08c411ecb806d","fingerprint":"f6-4f7c344a4c25ac22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f7c344a4c25ac22"},"other_id":"n-05fe2267b7e2ee62","other":{"id":"n-05fe2267b7e2ee62","name":"theta_plus_one_squared of Triangle read by rows, A007318 rows repeated.","first_index":0,"fingerprint":"f6-4f7c344a4c25ac22","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","4"]],"row_sums_preview":["1","1","5","5","18","18","56","56"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4124f4242e7d4c67","steps":["(\u03b8+1)\u00b2"],"edges":["e-e247b8d27ee67ca0"],"text":"A152198 \u2192 (\u03b8+1)\u00b2"},"headline":"A152198 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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:  2;\n  2:  2,   8;\n  3:  4,  16;\n  4:  4,  32,   36;\n  5:  8,  64,   72;\n  6:  8,  96,  216,  128;\n  7: 16, 192,  432,  256;\n  8: 16, 256,  864, 1024, 400;\n  9: 32, 512, 1728, 2048, 800;\n","recurrence_sentence":"The row polynomials satisfy (64+(80*n+56)*x+(56*n^2+24*n+4)*x^2+(12*n^3+22*n^2-6*n-12)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = -(64*n-192+(136*n^2-608*n+200)*x+(80*n^3-532*n^2+64*n+220)*x^2+(16*n^4-162*n^3-160*n^2+142*n+132)*x^3+(n^5-21*n^4-59*n^3-11*n^2+62*n+20)*x^4-(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) + (32+(96*n-112)*x+(88*n^2-184*n-312)*x^2+(24*n^3-52*n^2-564*n-288)*x^3+(2*n^4-4*n^3-254*n^2-416*n-160)*x^4-(40*n^3+140*n^2+156*n+48)*x^5-(2*n^4+12*n^3+26*n^2+24*n+8)*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 - 192 + (136*n^2 - 608*n + 200)*x + (80*n^3 - 532*n^2 + 64*n + 220)*x^2 + (16*n^4 - 162*n^3 - 160*n^2 + 142*n + 132)*x^3 + (n^5 - 21*n^4 - 59*n^3 - 11*n^2 + 62*n + 20)*x^4 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] + (32 + (96*n - 112)*x + (88*n^2 - 184*n - 312)*x^2 + (24*n^3 - 52*n^2 - 564*n - 288)*x^3 + (2*n^4 - 4*n^3 - 254*n^2 - 416*n - 160)*x^4 - (40*n^3 + 140*n^2 + 156*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^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 - 192 + (136*n^2 - 608*n + 200)*x + (80*n^3 - 532*n^2 + 64*n + 220)*x^2 + (16*n^4 - 162*n^3 - 160*n^2 + 142*n + 132)*x^3 + (n^5 - 21*n^4 - 59*n^3 - 11*n^2 + 62*n + 20)*x^4 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] + (32 + (96*n - 112)*x + (88*n^2 - 184*n - 312)*x^2 + (24*n^3 - 52*n^2 - 564*n - 288)*x^3 + (2*n^4 - 4*n^3 - 254*n^2 - 416*n - 160)*x^4 - (40*n^3 + 140*n^2 + 156*n + 48)*x^5 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 56)*x + (56*n^2 + 24*n + 4)*x^2 + (12*n^3 + 22*n^2 - 6*n - 12)*x^3 + (n^4 + 2*n^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-b6aca576a689e9f0","steps":["(\u03b8+1)\u00b2"],"edges":["e-e6a7da242cad039d"],"text":"A274496 \u2192 (\u03b8+1)\u00b2"},"headline":"A274496 \u2192 (\u03b8+1)\u00b2","chain":null}