{"id":"n-cc3556722af3bd66","revision":0,"first_index":1,"fingerprint":"f6-98685e9cd7383975","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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"],["2","6"],["8","48","27"],["32","288","486","108"],["128","1536","4320","3456","405"],["512","7680","30240","43200","20250","1458"],["2048","36864","186624","387072","340200","104976","5103"],["8192","172032","1064448","2903040","3810240","2286144","500094","17496"],["32768","786432","5750784","19464192","34214400","31352832","13716864","2239488","59049"],["131072","3538944","29859840","120766464","266872320","332563968","226328256","75582720","9565938","196830"],["524288","15728640","150405120","707788800","1886976000","2988969984","2802159360","1478062080","389723400","39366000","649539"],["2097152","69206016","739639296","3970695168","12408422400","23911759872","28768836096","21136287744","8916871392","1905314400","157188438","2125764"],["8388608","301989888","3567255552","21516779520","77157826560","175432531968","258681765888","245949530112","145912440960","50438868480","8916871392","612220032","6908733"],["33554432","1308622848","16930308096","113359454208","458912563200","1203662094336","2106408665088","2470674825216","1910912560128","936721843200","270478432224","40236460992","2335151754","22320522"],["134217728","5637144576","79272345600","583444463616","2632337326080","7828342308864","15879080706048","22221454049280","21327387734016","13719372226560","5659241043456","1386610348032","176034516840","8749644624","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","83","914","9845","103340","1062887","10761686","107616809","1065406352","10460353211","101988443738","988503377693","9531996856244","91507169819855"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","6"]],"recurrence":{"denominator":{"coeffs":[["12","-18"],["-14","39"],["4","-27"],["0","6"]]},"terms":[{"coeff":{"coeffs":[["48","144","36"],["-56","-96","108"],["16","-36","-171"],["0","24","54"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","42","18"],["0","-8","-12","36"],["0","0","-12","-36"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184020,"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-4919d190a5334150"]},{"id":184021,"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-4919d190a5334150"]},{"id":184022,"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-4919d190a5334150"]},{"id":184023,"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-4919d190a5334150"]}],"region_exact":{},"incoming_edges":[{"id":"e-4919d190a5334150","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-cc3556722af3bd66","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":"1ce4c0c32fe7a9d0644eb0e659fe629e6c9e04155eaa23e87b698cee0f8e8fcf","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-831045384d9c95c9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:      2,       6;\n   3:      8,      48,       27;\n   4:     32,     288,      486,       108;\n   5:    128,    1536,     4320,      3456,       405;\n   6:    512,    7680,    30240,     43200,     20250,      1458;\n   7:   2048,   36864,   186624,    387072,    340200,    104976,      5103;\n   8:   8192,  172032,  1064448,   2903040,   3810240,   2286144,    500094,    17496;\n   9:  32768,  786432,  5750784,  19464192,  34214400,  31352832,  13716864,  2239488,   59049;\n  10: 131072, 3538944, 29859840, 120766464, 266872320, 332563968, 226328256, 75582720, 9565938, 196830;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12+(6*n^3-27*n^2+39*n-18)*x)*P(n,x) = (16*n^2-56*n+48+(24*n^3-36*n^2-96*n+144)*x+(54*n^3-171*n^2+108*n+36)*x^2)*P(n-1,x) - ((8*n-12)*x+(12*n^2+12*n-42)*x^2+(36*n^2-36*n-18)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 56*n + 48 + (24*n^3 - 36*n^2 - 96*n + 144)*x + (54*n^3 - 171*n^2 + 108*n + 36)*x^2)*P[n - 1, x] - ((8*n - 12)*x + (12*n^2 + 12*n - 42)*x^2 + (36*n^2 - 36*n - 18)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (6*n^3 - 27*n^2 + 39*n - 18)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = ((16*n^2 - 56*n + 48 + (24*n^3 - 36*n^2 - 96*n + 144)*x + (54*n^3 - 171*n^2 + 108*n + 36)*x^2)*v[i - 1] - ((8*n - 12)*x + (12*n^2 + 12*n - 42)*x^2 + (36*n^2 - 36*n - 18)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (6*n^3 - 27*n^2 + 39*n - 18)*x)); 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-f7935c36a3c0ec8c","steps":["basis A013610","\u03b8+1"],"edges":["e-3b5bae8e6d048f8c","e-4919d190a5334150"],"text":"A086645 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A086645 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}