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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":"f9da2cbbcde0f7bb9570529740d18c648b2030c93c06070f7a01ebd1236c84a5","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-87b7b64af43d2638","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","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-ea068e5dc433974c","kind":"fingerprint_related","node_a_id":"n-a43d8d44a6eb5699","node_b_id":"n-66407c1a4552c33b","fingerprint":"f6-d115b0993943a210","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d115b0993943a210"},"other_id":"n-a43d8d44a6eb5699","other":{"id":"n-a43d8d44a6eb5699","name":"theta_plus_one_squared of basis_map_A013610 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-d115b0993943a210","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","48","81"],["16","384","1944","3456","2025"]],"card_rows":[["1"],["4","48","81"],["16","384","1944","3456","2025"],["64","2304","19440","69120","121500","104976","35721"]],"row_sums_preview":["1","133","7825","353125","13890625","501953125","17119140625","559814453125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-cb7f9b19b94bd36e","e-693fa493adf4b12c"],"text":"A034870 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A034870 \u2192 basis A013610 \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:         8,          96,          162;\n  2:        64,        1536,         7776,         13824,           8100;\n  3:       512,       18432,       155520,        552960,         972000,          839808,           285768;\n  4:      4096,      196608,      2322432,      12386304,       36288000,        62705664,         64012032,         35831808,           8503056;\n  5:     32768,     1966080,     29859840,     212336640,      870912000,      2257403904,       3840721920,       4299816960,        3061100160,        1259712000,         228637728;\n  6:    262144,    18874368,    350355456,    3114270720,    16422912000,     56757583872,     135193411584,     227030335488,      269376814080,      221709312000,      120720720384,       39182082048,        5748065856;\n  7:   2097152,   176160768,   3864526848,   41221619712,   265686220800,   1147764473856,    3515028701184,    7870384963584,    13073754710016,    16140437913600,    14647447406592,     9508185243648,     4184591943168,     1119954511872,      137749507200;\n  8:  16777216,  1610612736,  40768634880,  507343011840,  3864526848000,  20033707180032,   74987278958592,  209876932362240,   448243018629120,   737848590336000,   937436634021888,   912785783390208,   669534710906880,   358385443799040,   132239526912000,    30091839012864,    3184768606464;\n  9: 134217728, 14495514624, 415840075776, 5914169966592, 51980009472000, 314375097286656, 1390673173413888, 4670715731116032, 12192210106712064, 25086852071424000, 40979372858671104, 53202371374743552, 54634032410001408, 43866378321002496, 26976863490048000, 12277470317248512, 3898156774311936, 771220920950784, 71607703822848;\n","recurrence_sentence":"The row polynomials satisfy (4+(36*n-24)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (32+(288*n+192)*x+(288*n^2+1152*n+432)*x^2+(864*n^2+1512*n+432)*x^3+(648*n^2+648*n+162)*x^4)*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[((32 + (288*n + 192)*x + (288*n^2 + 1152*n + 432)*x^2 + (864*n^2 + 1512*n + 432)*x^3 + (648*n^2 + 648*n + 162)*x^4)*P[n - 1, x])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)]];\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] = ((32 + (288*n + 192)*x + (288*n^2 + 1152*n + 432)*x^2 + (864*n^2 + 1512*n + 432)*x^3 + (648*n^2 + 648*n + 162)*x^4)*v[i - 1])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)); 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-336459bdce627a87","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-a142fb8711dcbce4","e-9fe5d9f1b7f869f1"],"text":"A139548 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A139548 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}