{"id":"n-bf7b34a29f09cf83","revision":0,"first_index":1,"fingerprint":"f6-c7722c8666024957","name":"theta_plus_one of reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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"],["1","4"],["1","8","9"],["1","12","30","16"],["1","16","63","80","25"],["1","20","108","224","175","36"],["1","24","165","480","630","336","49"],["1","28","234","880","1650","1512","588","64"],["1","32","315","1456","3575","4752","3234","960","81"],["1","36","408","2240","6825","12012","12012","6336","1485","100"],["1","40","513","3264","11900","26208","35035","27456","11583","2200","121"],["1","44","630","4560","19380","51408","86632","91520","57915","20020","3146","144"],["1","48","759","6160","29925","93024","189924","254592","218790","114400","33033","4368","169"],["1","52","900","8096","44275","158004","379848","620160","680238","486200","213928","52416","5915","196"],["1","56","1053","10400","63250","255024","706629","1364352","1831410","1679600","1016158","381888","80444","7840","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","18","59","185","564","1685","4957","14406","41455","118321","335400","945193","2650229","7398330"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","-4"],["-6","16"],["4","-20"],["0","8"]]},"terms":[{"coeff":{"coeffs":[["2","12","16"],["-6","-28","-8"],["4","4","-36"],["0","8","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","6","8"],["0","-2","-8","0"],["0","0","-4","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172896,"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-312241ae192a8718"]},{"id":172897,"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-312241ae192a8718"]},{"id":172898,"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-312241ae192a8718"]}],"region_exact":{},"incoming_edges":[{"id":"e-312241ae192a8718","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-bf7b34a29f09cf83","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":"4bc86a057ba52ef6d0c17c23ca9e5907ec2a8ee714c60996d9578d2f808e30c6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-697c605ecd7e51b4","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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":[{"id":"l-986557bcdc97c885","kind":"fingerprint_related","node_a_id":"n-bf7b34a29f09cf83","node_b_id":"n-09dc7996013fc8cf","fingerprint":"f6-c7722c8666024957","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7722c8666024957"},"other_id":"n-09dc7996013fc8cf","other":{"id":"n-09dc7996013fc8cf","name":"theta_plus_one of basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-c7722c8666024957","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","16"],["16","128","144"]],"card_rows":[["1"],["4","16"],["16","128","144"],["64","768","1920","1024"]],"row_sums_preview":["1","20","288","3776","47360","577536","6901760","81215488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013611","\u03b8+1"],"edges":["e-61d2480a3633cd97","e-85d75586dc0ad6f3"],"text":"A091044 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A091044 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-d3e84c473e5d3b6f","kind":"fingerprint_related","node_a_id":"n-68ad8eba8fb6df33","node_b_id":"n-bf7b34a29f09cf83","fingerprint":"f6-c7722c8666024957","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7722c8666024957"},"other_id":"n-68ad8eba8fb6df33","other":{"id":"n-68ad8eba8fb6df33","name":"theta_plus_one of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-c7722c8666024957","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["1","8","9"]],"card_rows":[["1"],["1","4"],["1","8","9"],["1","12","30","16"]],"row_sums_preview":["1","5","18","59","185","564","1685","4957"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["\u03b8+1"],"edges":["e-22126d71e196a99d"],"text":"A172431 \u2192 \u03b8+1"},"headline":"A172431 \u2192 \u03b8+1","chain":null}}],"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: 1,  4;\n   3: 1,  8,   9;\n   4: 1, 12,  30,   16;\n   5: 1, 16,  63,   80,   25;\n   6: 1, 20, 108,  224,  175,    36;\n   7: 1, 24, 165,  480,  630,   336,    49;\n   8: 1, 28, 234,  880, 1650,  1512,   588,   64;\n   9: 1, 32, 315, 1456, 3575,  4752,  3234,  960,   81;\n  10: 1, 36, 408, 2240, 6825, 12012, 12012, 6336, 1485, 100;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-6*n+2+(8*n^3-20*n^2+16*n-4)*x)*P(n,x) = (4*n^2-6*n+2+(8*n^3+4*n^2-28*n+12)*x+(24*n^3-36*n^2-8*n+16)*x^2)*P(n-1,x) - ((2*n-1)*x+(4*n^2+8*n-6)*x^2+(16*n^2-8)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*P[n - 1, x] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^2 - 6*n + 2 + (8*n^3 + 4*n^2 - 28*n + 12)*x + (24*n^3 - 36*n^2 - 8*n + 16)*x^2)*v[i - 1] - ((2*n - 1)*x + (4*n^2 + 8*n - 6)*x^2 + (16*n^2 - 8)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (8*n^3 - 20*n^2 + 16*n - 4)*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-976cdb6e4a693ec8","steps":["reverse","\u03b8+1"],"edges":["e-32f4efbf40925a37","e-312241ae192a8718"],"text":"A078812 \u2192 reverse \u2192 \u03b8+1"},"headline":"A078812 \u2192 reverse \u2192 \u03b8+1","chain":null}