{"id":"n-41bbb784f9572762","revision":1,"first_index":0,"fingerprint":"f6-d6816b839be5a6fb","name":"theta_plus_one of Number triangle T(n,k)=C(n,k)C(2,n-k).","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"],["1","8","3"],["0","6","18","4"],["0","0","18","32","5"],["0","0","0","40","50","6"],["0","0","0","0","75","72","7"],["0","0","0","0","0","126","98","8"],["0","0","0","0","0","0","196","128","9"],["0","0","0","0","0","0","0","288","162","10"],["0","0","0","0","0","0","0","0","405","200","11"],["0","0","0","0","0","0","0","0","0","550","242","12"],["0","0","0","0","0","0","0","0","0","0","726","288","13"],["0","0","0","0","0","0","0","0","0","0","0","936","338","14"],["0","0","0","0","0","0","0","0","0","0","0","0","1183","392","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","12","28","55","96","154","232","333","460","616","804","1027","1288","1590"],"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"],["1","8","3"],["0","6","18","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160990,"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-ac5d36b14b8fced4"]},{"id":160991,"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-ac5d36b14b8fced4"]},{"id":160992,"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-ac5d36b14b8fced4"]}],"region_exact":{},"incoming_edges":[{"id":"e-ac5d36b14b8fced4","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-41bbb784f9572762","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":"b1bd9980640ab24c78cb4cbdeb733d43ae3ad3ab4553737210b5689443b5e712","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-1815d3e1c9c57d5b","role":"sequence","name":"Number triangle T(n,k)=C(n,k)C(2,n-k).","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-1e2d0eb3f9880ab3","kind":"oeis_match","node_a_id":"n-41bbb784f9572762","node_b_id":null,"fingerprint":null,"oeis_id":"A006000","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A006000","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A006000","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-41bbb784f9572762","oeis_id":"A006000","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":48,"oeis_terms_available":48,"term_count":200,"raw_size":2059}],"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: 1, 8,  3;\n  3: 0, 6, 18,  4;\n  4: 0, 0, 18, 32,  5;\n  5: 0, 0,  0, 40, 50,   6;\n  6: 0, 0,  0,  0, 75,  72,   7;\n  7: 0, 0,  0,  0,  0, 126,  98,   8;\n  8: 0, 0,  0,  0,  0,   0, 196, 128,   9;\n  9: 0, 0,  0,  0,  0,   0,   0, 288, 162, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^4*P(n-4,x) + 4*x^3*P(n-3,x) - 6*x^2*P(n-2,x) + 4*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 1 + 8*x + 3*x^2, P(3,x) = 6*x + 18*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 1 + 8*x + 3*x^2;\nP[3, x_] := 6*x + 18*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^4*P[n - 4, x] + 4*x^3*P[n - 3, x] - 6*x^2*P[n - 2, x] + 4*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; v[3] = 1 + 8*x + 3*x^2; v[4] = 6*x + 18*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^4*v[i - 4] + 4*x^3*v[i - 3] - 6*x^2*v[i - 2] + 4*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-6fea5aff4f44fa54","steps":["(1/2)D\u00b2+2D+1","\u03b8+1"],"edges":["e-0c9044372fd705ed","e-ac5d36b14b8fced4"],"text":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 \u03b8+1"},"headline":"A023531 \u2192 (1/2)D\u00b2+2D+1 \u2192 \u03b8+1","chain":null}