{"id":"n-8e8e81fc9d22b8ab","revision":1,"first_index":0,"fingerprint":"f6-7820744f6c385cb4","name":"theta_plus_one of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","6","3"],["1","12","33","24","5"],["1","18","90","180","150","54","7"],["1","24","174","576","975","864","406","96","9"],["1","30","285","1320","3425","5238","4795","2640","855","150","11"],["1","36","423","2520","8850","19548","27923","26064","15930","6300","1551","216","13"],["1","42","588","4284","19005","55188","108731","147864","139797","91980","41811","12852","2548","294","15"],["1","48","780","6720","36050","130032","328496","594240","778635","742800","516208","260064","93730","23520","3900","384","17"],["1","54","999","9936","62550","269892","834078","1893888","3206547","4081050","3919113","2840832","1549002","629748","187650","39744","5661","486","19"],["1","60","1245","14040","101475","509976","1866270","5113440","10670805","17136900","21305691","20564280","15413385","8948520","3999150","1359936","345015","63180","7885","600","21"],["1","66","1518","19140","156200","896346","3793251","12192048","30328155","59098050","90899556","110856276","107426748","82737270","50546925","24384096","9212181","2689038","593560","95700","10626","726","23"],["1","72","1818","25344","230505","1487376","7149142","26392608","76353354","175431960","322990866","479263392","575136393","559140624","440442090","280691136","144223002","59383368","19404814","4957920","968121","139392","13938","864","25"],["1","78","2145","32760","328575","2355210","12677665","52887120","174620160","462933900","995237100","1746842760","2514458895","2976414210","2901298725","2329123680","1538093700","833281020","368642560","132217800","38032995","8635770","1511445","196560","17875","1014","27"],["1","84","2499","41496","455000","3587220","21380905","99493680","369414045","1111110000","2736343610","5559503040","9367887165","13138033860","15369643125","15014895840","12250313985","8339254560","4726411690","2222220000","861966105","273607620","70251545","14348880","2275000","269724","22491","1176","29"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","10","75","500","3125","18750","109375","625000","3515625","19531250","107421875","585937500","3173828125","17089843750","91552734375"],"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":[["1","0","-1"],["0","3","2"]]},"terms":[{"coeff":{"coeffs":[["1","6","11","6","1"],["0","3","11","9","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165541,"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-c9b2fef810d9f0c1"]},{"id":165542,"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-c9b2fef810d9f0c1"]},{"id":165543,"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-c9b2fef810d9f0c1"]}],"region_exact":{},"incoming_edges":[{"id":"e-c9b2fef810d9f0c1","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-8e8e81fc9d22b8ab","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":"867e1f6dcf08615068cdcdd275a248f8e1b780503e0c7b27602f8879adbefea2","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-b30e7a5b4198051a","role":"sequence","name":"Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 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-52374f8a8dd0e142","kind":"oeis_match","node_a_id":"n-8e8e81fc9d22b8ab","node_b_id":null,"fingerprint":null,"oeis_id":"A053464","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A053464","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A053464","relation":"match_row_sums"}],"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,  6,   3;\n  2: 1, 12,  33,   24,     5;\n  3: 1, 18,  90,  180,   150,     54,      7;\n  4: 1, 24, 174,  576,   975,    864,    406,      96,       9;\n  5: 1, 30, 285, 1320,  3425,   5238,   4795,    2640,     855,     150,      11;\n  6: 1, 36, 423, 2520,  8850,  19548,  27923,   26064,   15930,    6300,    1551,     216,      13;\n  7: 1, 42, 588, 4284, 19005,  55188, 108731,  147864,  139797,   91980,   41811,   12852,    2548,    294,     15;\n  8: 1, 48, 780, 6720, 36050, 130032, 328496,  594240,  778635,  742800,  516208,  260064,   93730,  23520,   3900,   384,   17;\n  9: 1, 54, 999, 9936, 62550, 269892, 834078, 1893888, 3206547, 4081050, 3919113, 2840832, 1549002, 629748, 187650, 39744, 5661, 486, 19;\n","recurrence_sentence":"The row polynomials satisfy (1+3*n*x+(2*n-1)*x^2)*P(n,x) = (1+(3*n+6)*x+(11*n+11)*x^2+(9*n+6)*x^3+(2*n+1)*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[((1 + (3*n + 6)*x + (11*n + 11)*x^2 + (9*n + 6)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(1 + 3*n*x + (2*n - 1)*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] = ((1 + (3*n + 6)*x + (11*n + 11)*x^2 + (9*n + 6)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(1 + 3*n*x + (2*n - 1)*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-b30e7a5b4198051a","steps":["\u03b8+1"],"edges":["e-c9b2fef810d9f0c1"],"text":"A272866 \u2192 \u03b8+1"},"headline":"A272866 \u2192 \u03b8+1","chain":null}
