{"id":"n-a74afe30a81c362f","revision":0,"first_index":0,"fingerprint":"f6-d744f831d9613171","name":"theta_plus_one of Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","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"],["1","2"],["2","10","9"],["5","42","84","48"],["14","168","540","660","275"],["42","660","2970","5720","5005","1638"],["132","2574","15015","40040","54600","37128","9996"],["429","10010","72072","247520","464100","488376","271320","62016"],["1430","38896","334152","1410864","3391500","4883760","4178328","1961256","389367"],["4862","151164","1511640","7596960","22383900","41186376","48050772","34610400","14060475","2466750"],["16796","587860","6715170","39225120","137287920","308897820","459308850","449935200","279729450","100150050","15737865"],["58786","2288132","29418840","196125600","796760250","2125943820","3858194340","4807202400","4056077025","2217608250","709634640","100975680"],["208012","8914800","127481640","956112300","4429049625","13700526840","29444114700","44706982320","47900338200","35481732000","17308914480","5006710800","650872404"],["742900","34767720","547591590","4566842280","23785636875","83825591850","208632584160","374687089920","488583449640","458948490000","302906003400","133378775712","35197176924","4211628008"],["2674440","135727830","2335317075","21450319800","124186062000","491776805520","1393367615640","2895309331200","4460979322800","5094328239000","4258858407804","2534196738528","1016807333360","246681069040","27341497800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","21","179","1657","16035","159485","1615843","16589553","172023299","1797592101","18900287763","199721958121","2119306370979","22567756461837"],"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"],["2","0"],["4","4"],["0","8"]]},"terms":[{"coeff":{"coeffs":[["0","6","6"],["-6","-28","-24"],["12","18","6"],["0","28","36"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-9","-6"],["1","8","19","12"],["0","2","20","18"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180916,"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-6776e7877233c74b"]},{"id":180917,"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-6776e7877233c74b"]},{"id":180918,"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-6776e7877233c74b"]}],"region_exact":{},"incoming_edges":[{"id":"e-6776e7877233c74b","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-a74afe30a81c362f","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":"8792bf8d62ff98fc6c4a46aafaef37686600d84300932818e17c71d82c2ac335","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-9ed87ca196739b1c","role":"sequence","name":"Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","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-0b337f8648a881a6","kind":"fingerprint_related","node_a_id":"n-a74afe30a81c362f","node_b_id":"n-313fdaaf3d4e08f3","fingerprint":"f6-d744f831d9613171","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d744f831d9613171"},"other_id":"n-313fdaaf3d4e08f3","other":{"id":"n-313fdaaf3d4e08f3","name":"theta_plus_one of Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-d744f831d9613171","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","30","27"]],"card_rows":[["1"],["2","4"],["6","30","27"],["20","168","336","192"]],"row_sums_preview":["1","6","63","716","8285","96210","1116395","12926744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eb78d61c7c9e8c34","steps":["\u03b8+1"],"edges":["e-b7d89e5aedaaf812"],"text":"A330798 \u2192 \u03b8+1"},"headline":"A330798 \u2192 \u03b8+1","chain":null}}],"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,      2;\n  2:    2,     10,       9;\n  3:    5,     42,      84,      48;\n  4:   14,    168,     540,     660,      275;\n  5:   42,    660,    2970,    5720,     5005,     1638;\n  6:  132,   2574,   15015,   40040,    54600,    37128,     9996;\n  7:  429,  10010,   72072,  247520,   464100,   488376,   271320,    62016;\n  8: 1430,  38896,  334152, 1410864,  3391500,  4883760,  4178328,  1961256,   389367;\n  9: 4862, 151164, 1511640, 7596960, 22383900, 41186376, 48050772, 34610400, 14060475, 2466750;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n+(8*n^3+4*n^2)*x)*P(n,x) = (12*n^2-6*n+(28*n^3+18*n^2-28*n+6)*x+(36*n^3+6*n^2-24*n+6)*x^2)*P(n-1,x) + (n+(2*n^2+8*n-3)*x+(20*n^2+19*n-9)*x^2+(18*n^2+12*n-6)*x^3)*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[((12*n^2 - 6*n + (28*n^3 + 18*n^2 - 28*n + 6)*x + (36*n^3 + 6*n^2 - 24*n + 6)*x^2)*P[n - 1, x] + (n + (2*n^2 + 8*n - 3)*x + (20*n^2 + 19*n - 9)*x^2 + (18*n^2 + 12*n - 6)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*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] = ((12*n^2 - 6*n + (28*n^3 + 18*n^2 - 28*n + 6)*x + (36*n^3 + 6*n^2 - 24*n + 6)*x^2)*v[i - 1] + (n + (2*n^2 + 8*n - 3)*x + (20*n^2 + 19*n - 9)*x^2 + (18*n^2 + 12*n - 6)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (8*n^3 + 4*n^2)*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-9ed87ca196739b1c","steps":["\u03b8+1"],"edges":["e-6776e7877233c74b"],"text":"A104978 \u2192 \u03b8+1"},"headline":"A104978 \u2192 \u03b8+1","chain":null}