{"id":"n-e9930db7b976d5ff","revision":0,"first_index":0,"fingerprint":"f6-82e87d4c95b4bd9c","name":"theta_plus_one of basis_map_A013611 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","8"],["4","24"],["10","96","144"],["26","320","720"],["76","1200","4320","3840"],["232","4368","20160","26880"],["764","17024","100800","215040","134400"],["2620","66816","471744","1290240","1209600"],["9496","275040","2298240","8064000","12096000","5806080"],["35696","1152800","11024640","46126080","88704000","63866880"],["140152","5013888","54457920","269660160","665280000","766402560","298045440"],["568504","22274304","269755200","1528750080","4497292800","6642155520","3874590720"],["2390480","102030656","1368791424","8810081280","30673843200","58118860800","54244270080","17712414720"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","10","28","250","1066","9436","51640","468028","3041020","28548856","210910096","2059000120","16835387128","171032682640"],"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":[["4","16","0"],["0","8","8"],["0","0","8"]]},"terms":[{"coeff":{"coeffs":[["0","-24","-128","-64"],["4","36","64","-32"],["0","12","72","64"],["0","0","8","32"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-16","-88","-192","-128"],["0","-4","-44","-160","-192"],["0","0","-4","-32","-64"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176943,"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-536d091954c3b8cf"]},{"id":176944,"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-536d091954c3b8cf"]},{"id":176945,"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-536d091954c3b8cf"]}],"region_exact":{},"incoming_edges":[{"id":"e-536d091954c3b8cf","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-e9930db7b976d5ff","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":"3ebea389d4533ac113ff7cc1c59de484fed51ae19885d1fb3dc1b98847c7f333","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-378328f3fde82415","role":"sequence","name":"basis_map_A013611 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","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:    1;\n  2:    2,     8;\n  3:    4,    24;\n  4:   10,    96,    144;\n  5:   26,   320,    720;\n  6:   76,  1200,   4320,    3840;\n  7:  232,  4368,  20160,   26880;\n  8:  764, 17024, 100800,  215040,  134400;\n  9: 2620, 66816, 471744, 1290240, 1209600;\n","recurrence_sentence":"The row polynomials satisfy (4+(8*n+16)*x+(8*n^2+8*n)*x^2)*P(n,x) = (4*n+(12*n^2+36*n-24)*x+(8*n^3+72*n^2+64*n-128)*x^2+(32*n^3+64*n^2-32*n-64)*x^3)*P(n-1,x) - (1+(4*n+16)*x+(4*n^2+44*n+88)*x^2+(32*n^2+160*n+192)*x^3+(64*n^2+192*n+128)*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[((4*n + (12*n^2 + 36*n - 24)*x + (8*n^3 + 72*n^2 + 64*n - 128)*x^2 + (32*n^3 + 64*n^2 - 32*n - 64)*x^3)*P[n - 1, x] - (1 + (4*n + 16)*x + (4*n^2 + 44*n + 88)*x^2 + (32*n^2 + 160*n + 192)*x^3 + (64*n^2 + 192*n + 128)*x^4)*D[P[n - 1, x], x])/(4 + (8*n + 16)*x + (8*n^2 + 8*n)*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] = ((4*n + (12*n^2 + 36*n - 24)*x + (8*n^3 + 72*n^2 + 64*n - 128)*x^2 + (32*n^3 + 64*n^2 - 32*n - 64)*x^3)*v[i - 1] - (1 + (4*n + 16)*x + (4*n^2 + 44*n + 88)*x^2 + (32*n^2 + 160*n + 192)*x^3 + (64*n^2 + 192*n + 128)*x^4)*deriv(v[i - 1], x))/(4 + (8*n + 16)*x + (8*n^2 + 8*n)*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-97b51734d39ede6c","steps":["basis A013611","\u03b8+1"],"edges":["e-ad55a1201a8e5f74","e-536d091954c3b8cf"],"text":"A100861 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A100861 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}