{"id":"n-8802fa67ca7b6a9f","revision":0,"first_index":0,"fingerprint":"f6-959faa625dfaa291","name":"theta_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["8","12","3"],["48","88","36","4"],["384","800","420","80","5"],["3840","8768","5400","1360","150","6"],["46080","112896","77952","23520","3500","252","7"],["645120","1672704","1260672","433216","78400","7728","392","8"],["10321920","28053504","22679808","8612352","1795920","217728","15288","576","9"],["185794560","525606912","450316800","185262080","43092000","6074208","529200","27840","810","10"],["3715891200","10883727360","9794746368","4305664000","1093417600","173194560","17670576","1161600","47520","1100","11"],["81749606400","246873784320","231810011136","107784269824","29437267200","5122381440","590812992","45750144","2352240","77000","1452","12"],["1961990553600","6088470036480","5933750943744","2895902490624","841224750080","158261875200","20155623488","1773218304","107922672","4461600","119548","1872","13"],["51011754393600","162224202055680","163410229592064","83205132681216","25491721615360","5124278455296","708685065088","69138674176","4800860064","235915680","8016008","179088","2366","14"],["1428329123020800","4644301166346240","4818822731661312","2547624021196800","817774621081600","174069862686720","25821506686976","2745808665600","212205090240","11939928000","483955472","13759200","260260","2940","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","23","176","1689","19524","264207","4098240","71697105","1396704420","29985521895","703416314160","17901641997225","491250187505700","14459713484342175"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","4","-1"],["8","-4","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184578,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-6b6fe28ce0cbc59d"]},{"id":184579,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-6b6fe28ce0cbc59d"]},{"id":184580,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-6b6fe28ce0cbc59d"]},{"id":184581,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-6b6fe28ce0cbc59d"]}],"region_exact":{},"incoming_edges":[{"id":"e-6b6fe28ce0cbc59d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8802fa67ca7b6a9f","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"8e3c4b6a2d945b8394397cc529e38ac3d027731be14e56e1edd00fe2610b37cd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fdb0bda32eba4618","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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-0348b397829aa9fd","kind":"fingerprint_related","node_a_id":"n-8802fa67ca7b6a9f","node_b_id":"n-df9ff482d15e6222","fingerprint":"f6-959faa625dfaa291","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-959faa625dfaa291"},"other_id":"n-df9ff482d15e6222","other":{"id":"n-df9ff482d15e6222","name":"derivative of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-959faa625dfaa291","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","6"]],"card_rows":[["1"],["6","6"],["120","180","45"],["5040","9240","3780","420"]],"row_sums_preview":["0","1","12","345","18480","1596105","202951980","35703612945"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["D"],"edges":["e-5f410c7b8b40e256"],"text":"A204420 \u2192 D"},"headline":"A204420 \u2192 D","chain":null}},{"id":"l-d2b377a7ee71129d","kind":"fingerprint_related","node_a_id":"n-8802fa67ca7b6a9f","node_b_id":"n-b8a0748f602aaf4f","fingerprint":"f6-959faa625dfaa291","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-959faa625dfaa291"},"other_id":"n-b8a0748f602aaf4f","other":{"id":"n-b8a0748f602aaf4f","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-959faa625dfaa291","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["32","48","12"]],"card_rows":[["1"],["4","4"],["32","48","12"],["384","704","288","32"]],"row_sums_preview":["1","8","92","1408","27024","624768","16909248","524574720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013609","\u03b8+1"],"edges":["e-8d6d917622b49386","e-46750f40ecde5238"],"text":"A225471 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A225471 \u2192 basis A013609 \u2192 \u03b8+1","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,         2;\n  2:         8,        12,         3;\n  3:        48,        88,        36,         4;\n  4:       384,       800,       420,        80,        5;\n  5:      3840,      8768,      5400,      1360,      150,       6;\n  6:     46080,    112896,     77952,     23520,     3500,     252,      7;\n  7:    645120,   1672704,   1260672,    433216,    78400,    7728,    392,     8;\n  8:  10321920,  28053504,  22679808,   8612352,  1795920,  217728,  15288,   576,   9;\n  9: 185794560, 525606912, 450316800, 185262080, 43092000, 6074208, 529200, 27840, 810, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(4*n-4)*x+x^2)*P(n-2,x) + (4*n-2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (4*n - 4)*x + x^2)*P[n - 2, x] + (4*n - 2 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (4*n - 4)*x + x^2)*v[i - 2] + (4*n - 2 + 2*x)*v[i - 1]); 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-4f6e747b794bcc2e","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-29cb0c45f74a7f15","e-6b6fe28ce0cbc59d"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}