{"id":"n-fdeb1554f3b02f13","revision":0,"first_index":0,"fingerprint":"f6-a546e680aefa4794","name":"basis_map_A008277 of Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).","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"],["2","2"],["6","12","3"],["16","56","32","4"],["40","230","230","65","5"],["96","876","1380","666","114","6"],["224","3178","7462","5495","1575","182","7"],["512","11152","37744","39928","17080","3256","272","8"],["1152","38214","182394","267057","160209","44856","6111","387","9"],["2560","128660","853100","1686310","1364790","526890","104160","10660","530","10"],["5632","427394","3895166","10213115","10862555","5558322","1496187","220110","17556","704","11"],["12288","1404888","17463336","59957172","82241940","54295428","19121256","3793284","431640","27600","912","12"],["26624","4579198","77196418","343686629","599526213","500835764","224537599","57699213","8785491","796510","41756","1157","13"],["57344","14823004","337469860","1933754914","4243985746","4421174758","2473839368","801048248","156750594","18896878","1397396","61166","1442","14"],["122880","47706810","1462181190","10720976175","29354645775","37705938270","25941310395","10374936000","2537736630","390660270","38213175","2349165","87165","1770","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","21","108","570","3138","18123","109952","700389","4677670","32696752","238749756","1817712585","14403260732","118576865685"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["6","12","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","10","2"],["-4","-5","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","10","4"],["0","-5","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2"],["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-4"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196144,"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-f9832e6a5c7b3771"]},{"id":196145,"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-f9832e6a5c7b3771"]},{"id":196146,"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-f9832e6a5c7b3771"]}],"region_exact":{},"incoming_edges":[{"id":"e-f9832e6a5c7b3771","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-fdeb1554f3b02f13","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e4ae65f9a24b77e8f6bd206ca406a0d677bf2ad2df6688b38a4e44c80aacfaa1","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-a88a7983f557eac4","role":"sequence","name":"Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).","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-88c66bc1889e9bc4","kind":"fingerprint_related","node_a_id":"n-5ecf178bc343139f","node_b_id":"n-fdeb1554f3b02f13","fingerprint":"f6-a546e680aefa4794","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a546e680aefa4794"},"other_id":"n-5ecf178bc343139f","other":{"id":"n-5ecf178bc343139f","name":"basis_map_A008277 of Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-a546e680aefa4794","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["2","4","1"]],"card_rows":[["1"],["1","1"],["2","4","1"],["4","14","8","1"]],"row_sums_preview":["1","2","7","27","114","523","2589","13744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3783f6df357ea675","steps":["basis A008277"],"edges":["e-1354b132e491ca6c"],"text":"A097805 \u2192 basis A008277"},"headline":"A097805 \u2192 basis A008277","chain":null}},{"id":"l-b8ca2f431068cace","kind":"fingerprint_related","node_a_id":"n-fdeb1554f3b02f13","node_b_id":"n-bd5fa5857f8c49a3","fingerprint":"f6-a546e680aefa4794","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a546e680aefa4794"},"other_id":"n-bd5fa5857f8c49a3","other":{"id":"n-bd5fa5857f8c49a3","name":"basis_map_A008277 of Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357368.","first_index":0,"fingerprint":"f6-a546e680aefa4794","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["8","16","4"]],"card_rows":[["1"],["2","2"],["8","16","4"],["32","112","64","8"]],"row_sums_preview":["1","4","28","216","1824","16736","165696","1759232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff74e25052457f18","steps":["basis A008277"],"edges":["e-b992ac0a9b57400a"],"text":"A380114 \u2192 basis A008277"},"headline":"A380114 \u2192 basis A008277","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:    2,      2;\n  2:    6,     12,      3;\n  3:   16,     56,     32,       4;\n  4:   40,    230,    230,      65,       5;\n  5:   96,    876,   1380,     666,     114,      6;\n  6:  224,   3178,   7462,    5495,    1575,    182,      7;\n  7:  512,  11152,  37744,   39928,   17080,   3256,    272,     8;\n  8: 1152,  38214, 182394,  267057,  160209,  44856,   6111,   387,   9;\n  9: 2560, 128660, 853100, 1686310, 1364790, 526890, 104160, 10660, 530, 10;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(5*n-10)*x+(n-2)*x^2)*P(n-2,x) - ((5*n-10)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) + (2*n-4)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 6 + 12*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*P[n - 2, x] - ((5*n - 10)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x] + (2*n - 4)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 6 + 12*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*v[i - 2] - ((5*n - 10)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 8 + (2*n - 4)*x)*v[i - 1] + (2*n - 4)*x*deriv(v[i - 1], x))/(n - 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-a88a7983f557eac4","steps":["basis A008277"],"edges":["e-f9832e6a5c7b3771"],"text":"A127952 \u2192 basis A008277"},"headline":"A127952 \u2192 basis A008277","chain":null}
