{"id":"n-b92b7c60e520404c","revision":0,"first_index":0,"fingerprint":"f6-1006d0233b977996","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","12"],["7","36"],["25","216","324"],["81","840","1620"],["331","4500","14580","12960"],["1303","20412","79380","90720"],["5937","111216","567000","1088640","680400"],["26785","562896","3306744","7620480","6123600"],["133651","3205980","22521240","68040000","91854000","44089920"],["669351","17678100","139316760","484989120","785862000","484989120"],["3609673","105851592","952176060","3963738240","8419950000","8729804160","3394923840"],["19674097","626512536","6205013100","28977886080","70929658800","88268019840","44134009920"],["113525595","3941762916","43346226924","231061390560","676312837200","1103350248000","926814208320","302633210880"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","15","43","565","2541","32371","191815","2453193","17640505","229844791","1913504451","25570053565","239160774373","3287573410395"],"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":[["24","63","0"],["0","36","27"],["0","0","27"]]},"terms":[{"coeff":{"coeffs":[["-24","-279","-756","-324"],["48","270","297","-162"],["0","108","459","324"],["0","0","54","162"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-16","-180","-720","-1188","-648"],["0","-48","-396","-1080","-972"],["0","0","-36","-216","-324"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180253,"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-7a1fbdf9f46fdec4"]},{"id":180254,"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-7a1fbdf9f46fdec4"]},{"id":180255,"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-7a1fbdf9f46fdec4"]}],"region_exact":{},"incoming_edges":[{"id":"e-7a1fbdf9f46fdec4","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-b92b7c60e520404c","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":"31a37168164f8537432ea92720956ab75bb3948ef82441be29a9fdb983fd99c2","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-b2d4a92113cb2b7b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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:     1;\n  2:     3,     12;\n  3:     7,     36;\n  4:    25,    216,     324;\n  5:    81,    840,    1620;\n  6:   331,   4500,   14580,   12960;\n  7:  1303,  20412,   79380,   90720;\n  8:  5937, 111216,  567000, 1088640,  680400;\n  9: 26785, 562896, 3306744, 7620480, 6123600;\n","recurrence_sentence":"The row polynomials satisfy (24+(36*n+63)*x+(27*n^2+27*n)*x^2)*P(n,x) = (48*n-24+(108*n^2+270*n-279)*x+(54*n^3+459*n^2+297*n-756)*x^2+(162*n^3+324*n^2-162*n-324)*x^3)*P(n-1,x) - (16+(48*n+180)*x+(36*n^2+396*n+720)*x^2+(216*n^2+1080*n+1188)*x^3+(324*n^2+972*n+648)*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[((48*n - 24 + (108*n^2 + 270*n - 279)*x + (54*n^3 + 459*n^2 + 297*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*P[n - 1, x] - (16 + (48*n + 180)*x + (36*n^2 + 396*n + 720)*x^2 + (216*n^2 + 1080*n + 1188)*x^3 + (324*n^2 + 972*n + 648)*x^4)*D[P[n - 1, x], x])/(24 + (36*n + 63)*x + (27*n^2 + 27*n)*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] = ((48*n - 24 + (108*n^2 + 270*n - 279)*x + (54*n^3 + 459*n^2 + 297*n - 756)*x^2 + (162*n^3 + 324*n^2 - 162*n - 324)*x^3)*v[i - 1] - (16 + (48*n + 180)*x + (36*n^2 + 396*n + 720)*x^2 + (216*n^2 + 1080*n + 1188)*x^3 + (324*n^2 + 972*n + 648)*x^4)*deriv(v[i - 1], x))/(24 + (36*n + 63)*x + (27*n^2 + 27*n)*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-c914b69838ac70ec","steps":["basis A013610","\u03b8+1"],"edges":["e-3c0217934a464502","e-7a1fbdf9f46fdec4"],"text":"A059344 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A059344 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}