{"id":"n-ae1dd42a4cd258d9","revision":0,"first_index":1,"fingerprint":"f6-ccc00fc710735267","name":"one_plus_derivative of basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["11","8"],["79","192","64"],["673","2488","2496","512"],["6897","32320","50048","28672","4096"],["83427","458312","914560","814080","307200","32768"],["1167711","7209728","17096896","19824640","11653120","3145728","262144"],["18596745","125748216","338839872","471500288","363704320","152993792","31195136","2097152"],["332360865","2419051392","7202977536","11484225536","10708475904","5941755904","1886388224","301989888","16777216"],["6590531835","51004074888","164693584128","292203845632","314208362496","211920027648","89071288320","22171090944","2868903936","134217728"],["143639001135","1171296249408","4046870954304","7832921292800","9425641103360","7368793915392","3780949573632","1249231503360","250903265280","26843545600","1073741824"],["3413691510705","29128043269944","106614028179648","221935271168512","292768179855360","257305401819136","153667581247488","62193093574656","16612212080640","2753476689920","248034361344","8589934592"],["87872160913425","780290272774080","3002497394260608","6650890631925760","9476158051250176","9166915294986240","6177850813054976","2920114072584192","958049883783168","211478820945920","29457033199616","2267742732288","68719476736"],["2435795173307475","22410545110869960","90111047688994176","210666920646853632","320596950095319040","336171118725398528","250217324480561152","133886283083677696","51445823937970176","13979531077484544","2595833136545792","308430191460352","20547123544064","549755813888"],["72345724399331775","687170866614733440","2873257575804748224","7043607596795707392","11350636130101538816","12753095321519128576","10320673305086656512","6116831755342184448","2667856573195354112","851054012437364736","194890932473036800","30892153571901440","3170510498168832","184717953466368","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","335","6169","122033","2610347","60359967","1504675521","40294002465","1154865927555","35298164146095","1146647603691945","39463910891887185","1434846700227800475","54965671573839832575"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["11","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","48","-64"],["12","-32","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","16"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190012,"position":0,"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-cabfe833db9d74d5"]},{"id":190013,"position":1,"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-cabfe833db9d74d5"]},{"id":190014,"position":2,"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-cabfe833db9d74d5"]},{"id":190015,"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-cabfe833db9d74d5"]}],"region_exact":{},"incoming_edges":[{"id":"e-cabfe833db9d74d5","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-ae1dd42a4cd258d9","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":"5b181c31a36415edc4b1f7d26009241e559f26210978de0e0d4eaf6178da4bf3","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-f4f909be2b0f8048","role":"sequence","name":"basis_map_A013611 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:          1;\n   2:         11,           8;\n   3:         79,         192,           64;\n   4:        673,        2488,         2496,          512;\n   5:       6897,       32320,        50048,        28672,         4096;\n   6:      83427,      458312,       914560,       814080,       307200,        32768;\n   7:    1167711,     7209728,     17096896,     19824640,     11653120,      3145728,      262144;\n   8:   18596745,   125748216,    338839872,    471500288,    363704320,    152993792,    31195136,     2097152;\n   9:  332360865,  2419051392,   7202977536,  11484225536,  10708475904,   5941755904,  1886388224,   301989888,   16777216;\n  10: 6590531835, 51004074888, 164693584128, 292203845632, 314208362496, 211920027648, 89071288320, 22171090944, 2868903936, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(32*n-48)*x+64*x^2)*P(n-2,x) + (4*n-4+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 11 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 11 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*P[n - 2, x] + (4*n - 4 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 11 + 8*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (32*n - 48)*x + 64*x^2)*v[i - 2] + (4*n - 4 + 16*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013611","D+1"],"edges":["e-1182ea8561b370d4","e-cabfe833db9d74d5"],"text":"A161198 \u2192 basis A013611 \u2192 D+1"},"headline":"A161198 \u2192 basis A013611 \u2192 D+1","chain":null}