{"id":"n-e2eb2d94d1c8ba7b","revision":0,"first_index":0,"fingerprint":"f6-5bb9b91dbe4d6d19","name":"theta_plus_one of basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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"],["5","2"],["22","20","3"],["95","138","45","4"],["409","820","423","80","5"],["1760","4504","3210","952","125","6"],["7573","23580","21537","8800","1800","180","7"],["32585","119534","133425","69860","19625","3042","245","8"],["140206","592100","781815","500800","180050","38220","4753","320","9"],["603275","2882310","4396950","3336840","1467375","398034","67620","7008","405","10"],["2595757","13841800","23962770","21044400","10967775","3636360","788214","111360","9882","500","11"],["11168960","65753584","127385700","127161840","76742250","30149028","7980630","1436592","173475","13450","605","12"],["48057529","309580440","663670566","742523600","509760225","231926760","72712374","15969600","2453895","258500","17787","720","13"],["206780765","1446756506","3400566390","4216026568","3246728875","1680898986","610201200","158638080","29714850","3978700","371470","22968","845","14"],["889731238","6718602740","17181955011","23386650560","19974396910","11604789300","4793917695","1442654400","319680405","52134500","6180559","517920","29068","980","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","45","282","1737","10557","63477","378324","2238273","13159827","76958829","447966126","2596932009","15000686217","86371241301"],"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":[["0","0","0"],["-52","-10","0"],["0","0","0"],["0","0","2"]]},"terms":[{"coeff":{"coeffs":[["-130","-64","-10","0"],["-130","-77","-10","1"],["0","-13","5","2"],["0","0","5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-169","-195","-63","-5","0"],["0","0","13","10","1"],["0","0","13","10","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179875,"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-75ae1a77686f9d0e"]},{"id":179876,"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-75ae1a77686f9d0e"]},{"id":179877,"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-75ae1a77686f9d0e"]},{"id":179878,"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-75ae1a77686f9d0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-75ae1a77686f9d0e","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-e2eb2d94d1c8ba7b","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":"b748b5fb1a7f8afcef9e1f8d01fa94449c240b9187bc72cc8e167fdc0dd80d60","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-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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":[],"oeis":[],"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:      5,       2;\n  2:     22,      20,       3;\n  3:     95,     138,      45,       4;\n  4:    409,     820,     423,      80,       5;\n  5:   1760,    4504,    3210,     952,     125,      6;\n  6:   7573,   23580,   21537,    8800,    1800,    180,     7;\n  7:  32585,  119534,  133425,   69860,   19625,   3042,   245,    8;\n  8: 140206,  592100,  781815,  500800,  180050,  38220,  4753,  320,   9;\n  9: 603275, 2882310, 4396950, 3336840, 1467375, 398034, 67620, 7008, 405, 10;\n","recurrence_sentence":"The row polynomials satisfy -(52*n+10*n*x-2*n^3*x^2)*P(n,x) = -(130*n+130+(13*n^2+77*n+64)*x-(5*n^3+5*n^2-10*n-10)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (169+195*x-(13*n^2+13*n-63)*x^2-(10*n^2+10*n-5)*x^3-(n^2+n)*x^4)*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[(-(130*n + 130 + (13*n^2 + 77*n + 64)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (169 + 195*x - (13*n^2 + 13*n - 63)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-52*n - 10*n*x + 2*n^3*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] = (-(130*n + 130 + (13*n^2 + 77*n + 64)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (169 + 195*x - (13*n^2 + 13*n - 63)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-52*n - 10*n*x + 2*n^3*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-6466a8550c08a52f","steps":["basis A007318","\u03b8+1"],"edges":["e-235ea1c8ecbb7de6","e-75ae1a77686f9d0e"],"text":"A116414 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A116414 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}
