{"id":"n-db31c2ee901d3a4d","revision":1,"first_index":0,"fingerprint":"f6-3fd7550e3b6b8595","name":"theta_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","2"],["1","6","3"],["1","12","15","4"],["1","20","45","28","5"],["1","30","105","112","45","6"],["1","42","210","336","225","66","7"],["1","56","378","840","825","396","91","8"],["1","72","630","1848","2475","1716","637","120","9"],["1","90","990","3696","6435","6006","3185","960","153","10"],["1","110","1485","6864","15015","18018","12740","5440","1377","190","11"],["1","132","2145","12012","32175","48048","43316","24480","8721","1900","231","12"],["1","156","3003","20020","64350","116688","129948","93024","43605","13300","2541","276","13"],["1","182","4095","32032","121550","262548","352716","310080","183141","73150","19481","3312","325","14"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","3","10","32","99","299","887","2595","7508","21526","61251","173173","486925","1362627"],"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":[["18","-6"],["-12","13"],["0","-9"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["18","6","0"],["-12","1","2"],["0","-9","-3"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-24","-6","0"],["0","0","-8","-2"],["0","0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161222,"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-4a7114805c9287a9"]},{"id":161223,"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-4a7114805c9287a9"]},{"id":161224,"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-4a7114805c9287a9"]},{"id":161225,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-4a7114805c9287a9","e-fbf52a4d513a6632"]},{"id":161226,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-fbf52a4d513a6632"]},{"id":161227,"position":5,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-fbf52a4d513a6632"]},{"id":161228,"position":6,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-fbf52a4d513a6632"]}],"region_exact":{},"incoming_edges":[{"id":"e-4a7114805c9287a9","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-db31c2ee901d3a4d","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":"2ac1055e5848d709bff5a336b4b0087ae8c7ab486874bd2df240caeb7f22bf26","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-01fe968f3e05a927","role":"sequence","name":"Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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]"}]},{"id":"e-fbf52a4d513a6632","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-db31c2ee901d3a4d","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":"4ae391c0b240e0083122d3460b12203e435536c77935359fb386be019a607b6f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0eba401b117f16a9","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4eb27238b4709c46","kind":"oeis_match","node_a_id":"n-db31c2ee901d3a4d","node_b_id":null,"fingerprint":null,"oeis_id":"A038731","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A038731","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038731","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-db31c2ee901d3a4d","oeis_id":"A038731","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":9588}],"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: 1,  2;\n  3: 1,  6,   3;\n  4: 1, 12,  15,    4;\n  5: 1, 20,  45,   28,    5;\n  6: 1, 30, 105,  112,   45,    6;\n  7: 1, 42, 210,  336,  225,   66,   7;\n  8: 1, 56, 378,  840,  825,  396,  91,   8;\n  9: 1, 72, 630, 1848, 2475, 1716, 637, 120, 9;\n","recurrence_sentence":"The row polynomials satisfy -(12*n-18-(2*n^3-9*n^2+13*n-6)*x)*P(n,x) = -(12*n-18-(2*n^3-9*n^2+n+6)*x-(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - (24*x-(4*n^2-8*n-6)*x^2-(n^2-2*n)*x^3)*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 - 18 - (2*n^3 - 9*n^2 + n + 6)*x - (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - (24*x - (4*n^2 - 8*n - 6)*x^2 - (n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(-12*n + 18 + (2*n^3 - 9*n^2 + 13*n - 6)*x)]];\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 - 18 - (2*n^3 - 9*n^2 + n + 6)*x - (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] - (24*x - (4*n^2 - 8*n - 6)*x^2 - (n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(-12*n + 18 + (2*n^3 - 9*n^2 + 13*n - 6)*x)); 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-01fe968f3e05a927","steps":["\u03b8+1"],"edges":["e-4a7114805c9287a9"],"text":"A165253 \u2192 \u03b8+1"},"headline":"A165253 \u2192 \u03b8+1","chain":null}
