{"id":"n-ff3909e492eb580d","revision":1,"first_index":0,"fingerprint":"f6-b144b587414c3e17","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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":true,"rows":[["1"],["5","3"],["18","21","6"],["56","96","54","10"],["160","360","300","110","15"],["432","1200","1320","720","195","21"],["1120","3696","5040","3640","1470","315","28"],["2816","10752","17472","15680","8400","2688","476","36"],["6912","29952","56448","60480","40320","17136","4536","684","45"],["16640","80640","172800","215040","171360","90720","31920","7200","945","55"],["39424","211200","506880","718080","665280","421344","184800","55440","10890","1265","66"],["92160","540672","1436160","2280960","2407680","1774080","931392","348480","91080","15840","1650","78"],["212992","1357824","3953664","6955520","8236800","6918912","4228224","1894464","617760","143000","22308","2106","91"],["487424","3354624","10622976","20500480","26906880","25369344","17681664","9225216","3603600","1041040","216216","30576","2639","105"],["1105920","8171520","27955200","58705920","84564480","88408320","69189120","41184000","18738720","6486480","1681680","316680","40950","3255","120"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","45","216","945","3888","15309","58320","216513","787320","2814669","9920232","34543665","119042784","406552365"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","3"],["18","21","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","12","6","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-12","-12","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163916,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-74760f8a95c660e1"]},{"id":163917,"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-74760f8a95c660e1"]},{"id":163918,"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-74760f8a95c660e1"]}],"region_exact":{},"incoming_edges":[{"id":"e-74760f8a95c660e1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ff3909e492eb580d","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"e9a44ecf57d62a5c24bf39c7aa88bb4ad9d0f8d3ec30166c7756c12dc13fbd60","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-787407680a56b31c","role":"sequence","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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-b56588a70d3f9a6f","kind":"oeis_match","node_a_id":"n-ff3909e492eb580d","node_b_id":null,"fingerprint":null,"oeis_id":"A383136","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A383136","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A383136","relation":"match_row_sums"}],"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,     3;\n  2:    18,    21,      6;\n  3:    56,    96,     54,     10;\n  4:   160,   360,    300,    110,     15;\n  5:   432,  1200,   1320,    720,    195,    21;\n  6:  1120,  3696,   5040,   3640,   1470,   315,    28;\n  7:  2816, 10752,  17472,  15680,   8400,  2688,   476,   36;\n  8:  6912, 29952,  56448,  60480,  40320, 17136,  4536,  684,  45;\n  9: 16640, 80640, 172800, 215040, 171360, 90720, 31920, 7200, 945, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+12*x+6*x^2+x^3)*P(n-3,x) - (12+12*x+3*x^2)*P(n-2,x) + (6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 3*x, P(2,x) = 18 + 21*x + 6*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 3*x;\nP[2, x_] := 18 + 21*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 12*x + 6*x^2 + x^3)*P[n - 3, x] - (12 + 12*x + 3*x^2)*P[n - 2, x] + (6 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 3*x; v[3] = 18 + 21*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 12*x + 6*x^2 + x^3)*v[i - 3] - (12 + 12*x + 3*x^2)*v[i - 2] + (6 + 3*x)*v[i - 1]); 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-c8c19eba9d68881f","steps":["(1/2+(1/2)n)\u03b8+1+n","x \u21a6 x+1"],"edges":["e-c94c2672fb190cd9","e-74760f8a95c660e1"],"text":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 (1/2+(1/2)n)\u03b8+1+n \u2192 x \u21a6 x+1","chain":null}
