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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":"fe8060af99a6343964555aa80dddea7fe97acc078800b3dd521b7a328d52123c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f9bed40d8798da1b","role":"sequence","name":"basis_map_A013609 of narayana","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-363622d8be72c64d","kind":"fingerprint_related","node_a_id":"n-0248f7b63cfca929","node_b_id":"n-842205b36b684b27","fingerprint":"f6-9d61137180e532ac","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9d61137180e532ac"},"other_id":"n-0248f7b63cfca929","other":{"id":"n-0248f7b63cfca929","name":"theta_plus_one_squared of basis_map_A013609 of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-9d61137180e532ac","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","8"],["5","40","36"]],"card_rows":[["1"],["2","8"],["5","40","36"],["14","168","324","128"]],"row_sums_preview":["1","10","81","634","4922","38084","294025","2265834"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-1b3fae304d93e9a3","e-a4caa1aec8ecacd9"],"text":"A001263 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A001263 \u2192 basis A013609 \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:     5,     40,      36;\n   4:    14,    168,     324,      128;\n   5:    42,    672,    2016,     1792,      400;\n   6:   132,   2640,   10800,    15360,     8000,     1152;\n   7:   429,  10296,   53460,   105600,    90000,    31104,     3136;\n   8:  1430,  40040,  252252,   640640,   770000,   443520,   109760,    8192;\n   9:  4862, 155584, 1153152,  3587584,  5605600,  4612608,  1931776,  360448,   20736;\n  10: 16796, 604656, 5155488, 19009536, 36691200, 39626496, 23971584, 7667712, 1119744, 51200;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+2*n^2+(6*n^4-3*n^3-8*n^2+n)*x+(2*n^5-2*n^4-2*n^3+2*n^2)*x^2)*P(n,x) = (8*n^3-4*n^2+(24*n^4-26*n^2-2*n)*x+(8*n^5+40*n^4-60*n^3-24*n^2+28*n+8)*x^2+(12*n^5-4*n^4-24*n^3+12*n^2+12*n-8)*x^3)*P(n-1,x) - (4*n^2*x+(12*n^3+8*n^2-5*n-1)*x^2+(4*n^4+28*n^3-4*n^2-12*n)*x^3+(8*n^4+8*n^3-8*n^2-4*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 4*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (8*n^5 + 40*n^4 - 60*n^3 - 24*n^2 + 28*n + 8)*x^2 + (12*n^5 - 4*n^4 - 24*n^3 + 12*n^2 + 12*n - 8)*x^3)*P[n - 1, x] - (4*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (4*n^4 + 28*n^3 - 4*n^2 - 12*n)*x^3 + (8*n^4 + 8*n^3 - 8*n^2 - 4*n + 4)*x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (2*n^5 - 2*n^4 - 2*n^3 + 2*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 4*n^2 + (24*n^4 - 26*n^2 - 2*n)*x + (8*n^5 + 40*n^4 - 60*n^3 - 24*n^2 + 28*n + 8)*x^2 + (12*n^5 - 4*n^4 - 24*n^3 + 12*n^2 + 12*n - 8)*x^3)*v[i - 1] - (4*n^2*x + (12*n^3 + 8*n^2 - 5*n - 1)*x^2 + (4*n^4 + 28*n^3 - 4*n^2 - 12*n)*x^3 + (8*n^4 + 8*n^3 - 8*n^2 - 4*n + 4)*x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n^2 + (6*n^4 - 3*n^3 - 8*n^2 + n)*x + (2*n^5 - 2*n^4 - 2*n^3 + 2*n^2)*x^2)); 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-2a05d8340c329b21","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-2f325d6b3b1beb16","e-40806c1052d95c00"],"text":"narayana \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"narayana \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}