{"id":"n-6bc6556496dfb205","revision":0,"first_index":0,"fingerprint":"f6-df250c54b8e590c9","name":"theta_plus_one of basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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"],["4","4"],["15","32","12"],["56","184","144","32"],["209","928","1116","512","80"],["780","4364","7104","4992","1600","192"],["2911","19648","40392","38400","18800","4608","448"],["10864","85872","213408","256320","169600","63360","12544","1024"],["40545","367296","1070856","1555968","1300400","655872","197568","32768","2304"],["151316","1545492","5171904","8823168","8921920","5681088","2308096","581632","82944","5120"],["564719","6419936","24253236","47528448","56445200","43481088","22290688","7569408","1638144","204800","11264"],["2107560","26393128","111100848","245965920","335680000","303711744","188310528","80646144","23500800","4454400","495616","24576"],["7865521","107582816","499329540","1232604160","1901189600","1976997888","1439612160","745439232","273818880","69836800","11770880","1179648","53248"],["29354524","435400220","2208965760","6015996160","10350588800","12167134848","10183133184","6191652864","2749012992","883379200","200228864","30375936","2768896","114688"],["109552575","1751436160","9642734160","28721955840","54541156000","71532954624","67682901888","47296905216","24653452032","9572597760","2732578816","557187072","76836864","6422528","245760"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","59","416","2845","19032","125207","812992","5223577","33272680","210406931","1322391264","8267280373","51448106936","318878917295"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-6","-2","0"],["0","0","0"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-12","-13","-4","0"],["-12","-16","-4","1"],["0","-3","2","2"],["0","0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","-18","-11","-2","0"],["0","0","3","4","1"],["0","0","3","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180099,"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-ea3b7928017c2ff9"]},{"id":180100,"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-ea3b7928017c2ff9"]},{"id":180101,"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-ea3b7928017c2ff9"]},{"id":180102,"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-ea3b7928017c2ff9"]}],"region_exact":{},"incoming_edges":[{"id":"e-ea3b7928017c2ff9","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-6bc6556496dfb205","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":"86cbaad50a79e89c26c59819cfedd88587688188a5cc6076790ffc232bb61c0e","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-7d7dad3bd149bf81","role":"sequence","name":"basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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":[],"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:      4,       4;\n  2:     15,      32,      12;\n  3:     56,     184,     144,      32;\n  4:    209,     928,    1116,     512,      80;\n  5:    780,    4364,    7104,    4992,    1600,     192;\n  6:   2911,   19648,   40392,   38400,   18800,    4608,     448;\n  7:  10864,   85872,  213408,  256320,  169600,   63360,   12544,   1024;\n  8:  40545,  367296, 1070856, 1555968, 1300400,  655872,  197568,  32768,  2304;\n  9: 151316, 1545492, 5171904, 8823168, 8921920, 5681088, 2308096, 581632, 82944, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(6*n+2*n*x-n^3*x^2)*P(n,x) = -(12*n+12+(3*n^2+16*n+13)*x-(2*n^3+2*n^2-4*n-4)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (9+18*x-(3*n^2+3*n-11)*x^2-(4*n^2+4*n-2)*x^3-(n^2+n)*x^4)*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[(-(12*n + 12 + (3*n^2 + 16*n + 13)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (9 + 18*x - (3*n^2 + 3*n - 11)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-6*n - 2*n*x + n^3*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] = (-(12*n + 12 + (3*n^2 + 16*n + 13)*x - (2*n^3 + 2*n^2 - 4*n - 4)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (9 + 18*x - (3*n^2 + 3*n - 11)*x^2 - (4*n^2 + 4*n - 2)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-6*n - 2*n*x + n^3*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-f287bed89e07609f","steps":["basis A013609","\u03b8+1"],"edges":["e-7797a63e71923f20","e-ea3b7928017c2ff9"],"text":"A125662 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A125662 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}