{"id":"n-98201fe962aff497","revision":1,"first_index":0,"fingerprint":"f6-76caa1d365acf455","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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":true,"rows":[["1"],["5","1"],["24","10","1"],["112","72","15","1"],["512","448","144","20","1"],["2304","2560","1120","240","25","1"],["10240","13824","7680","2240","360","30","1"],["45056","71680","48384","17920","3920","504","35","1"],["196608","360448","286720","129024","35840","6272","672","40","1"],["851968","1769472","1622016","860160","290304","64512","9408","864","45","1"],["3670016","8519680","8847360","5406720","2150400","580608","107520","13440","1080","50","1"],["15728640","40370176","46858240","32440320","14868480","4730880","1064448","168960","18480","1320","55","1"],["67108864","188743680","242221056","187432960","97320960","35684352","9461760","1824768","253440","24640","1584","60","1"],["285212672","872415232","1226833920","1049624576","609157120","253034496","77316096","17571840","2965248","366080","32032","1872","65","1"],["1207959552","3992977408","6106906624","5725224960","3673686016","1705639936","590413824","154632192","30750720","4612608","512512","40768","2184","70","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","35","200","1125","6250","34375","187500","1015625","5468750","29296875","156250000","830078125","4394531250","23193359375"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["3","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["16","8","1"],["4","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165792,"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-0ef8ac6724e5260c"]},{"id":165793,"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-0ef8ac6724e5260c"]},{"id":165794,"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-0ef8ac6724e5260c"]},{"id":165795,"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-0ef8ac6724e5260c"]}],"region_exact":{},"incoming_edges":[{"id":"e-0ef8ac6724e5260c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-98201fe962aff497","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"df725a791861512c91e95be3f40901753f905856d31055d9ffb0628b70e7356d","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-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","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-9452a1d7ce8d48a5","kind":"oeis_match","node_a_id":"n-98201fe962aff497","node_b_id":null,"fingerprint":null,"oeis_id":"A081105","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":"A081105","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c3564b10ba5058d2","kind":"oeis_match","node_a_id":"n-98201fe962aff497","node_b_id":null,"fingerprint":null,"oeis_id":"A079027","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A079027","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A079027","relation":"match_row_sums"},{"oeis_id":"A081105","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-98201fe962aff497","oeis_id":"A079027","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":15284},{"node_id":"n-98201fe962aff497","oeis_id":"A081105","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":15141}],"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:      5,       1;\n  2:     24,      10,       1;\n  3:    112,      72,      15,      1;\n  4:    512,     448,     144,     20,      1;\n  5:   2304,    2560,    1120,    240,     25,     1;\n  6:  10240,   13824,    7680,   2240,    360,    30,    1;\n  7:  45056,   71680,   48384,  17920,   3920,   504,   35,   1;\n  8: 196608,  360448,  286720, 129024,  35840,  6272,  672,  40,  1;\n  9: 851968, 1769472, 1622016, 860160, 290304, 64512, 9408, 864, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+3+x)*P(n,x) = (4*n+16+(n+8)*x+x^2)*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 + 16 + (n + 8)*x + x^2)*P[n - 1, x])/(n + 3 + 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] = ((4*n + 16 + (n + 8)*x + x^2)*v[i - 1])/(n + 3 + 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-3dcaebc50e901d48","steps":["D+1"],"edges":["e-0ef8ac6724e5260c"],"text":"A038231 \u2192 D+1"},"headline":"A038231 \u2192 D+1","chain":null}