{"id":"n-5b4d475ced54f59f","revision":0,"first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","name":"basis_map_A007318 of (3,1) Pascal triangle.","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"],["4","1"],["8","6","1"],["16","20","8","1"],["32","56","36","10","1"],["64","144","128","56","12","1"],["128","352","400","240","80","14","1"],["256","832","1152","880","400","108","16","1"],["512","1920","3136","2912","1680","616","140","18","1"],["1024","4352","8192","8960","6272","2912","896","176","20","1"],["2048","9728","20736","26112","21504","12096","4704","1248","216","22","1"],["4096","21504","51200","72960","69120","45696","21504","7200","1680","260","24","1"],["8192","47104","123904","197120","211200","160512","88704","35904","10560","2200","308","26","1"],["16384","102400","294912","518144","619520","532224","337920","160512","57024","14960","2816","360","28","1"],["32768","221184","692224","1331200","1757184","1683968","1208064","658944","274560","86944","20592","3536","416","30","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","15","45","135","405","1215","3645","10935","32805","98415","295245","885735","2657205","7971615"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195244,"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-bc413433eedeab55"]},{"id":195245,"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-bc413433eedeab55"]},{"id":195246,"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-bc413433eedeab55"]},{"id":195247,"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-bc413433eedeab55"]}],"region_exact":{},"incoming_edges":[{"id":"e-bc413433eedeab55","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5b4d475ced54f59f","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f9245e61e6e8a9e5a39ca084f4c34a8dadeb19da3f63bf005d636151d2e0a261","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-92128fbba2409faa","role":"sequence","name":"(3,1) Pascal triangle.","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-63abb7321322d859","kind":"fingerprint_related","node_a_id":"n-5b4d475ced54f59f","node_b_id":"n-f5f6d7397eb76ee7","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-f5f6d7397eb76ee7","other":{"id":"n-f5f6d7397eb76ee7","name":"basis_map_A013611 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-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","12","16"]],"card_rows":[["1"],["1","4"],["2","12","16"],["4","32","80","64"]],"row_sums_preview":["1","5","30","180","1080","6480","38880","233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3783f6df357ea675","steps":["basis A013611"],"edges":["e-ce077fbc42f0c51e"],"text":"A097805 \u2192 basis A013611"},"headline":"A097805 \u2192 basis A013611","chain":null}},{"id":"l-67fd74dd7ce21f9a","kind":"fingerprint_related","node_a_id":"n-5b4d475ced54f59f","node_b_id":"n-118ddec8f1e698aa","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-118ddec8f1e698aa","other":{"id":"n-118ddec8f1e698aa","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["6","36","48"]],"card_rows":[["1"],["2","8"],["6","36","48"],["16","128","320","256"]],"row_sums_preview":["1","10","90","720","5400","38880","272160","1866240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a88a7983f557eac4","steps":["basis A013611"],"edges":["e-584997da1613bc2e"],"text":"A127952 \u2192 basis A013611"},"headline":"A127952 \u2192 basis A013611","chain":null}},{"id":"l-96a738aed4cb4f28","kind":"fingerprint_related","node_a_id":"n-5b4d475ced54f59f","node_b_id":"n-acacc82d3699f634","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-acacc82d3699f634","other":{"id":"n-acacc82d3699f634","name":"basis_map_A013611 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-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["8","48","64"]],"card_rows":[["1"],["2","8"],["8","48","64"],["32","256","640","512"]],"row_sums_preview":["1","10","120","1440","17280","207360","2488320","29859840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff74e25052457f18","steps":["basis A013611"],"edges":["e-c477fbeecbbc62a9"],"text":"A380114 \u2192 basis A013611"},"headline":"A380114 \u2192 basis A013611","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:    4,    1;\n  2:    8,    6,    1;\n  3:   16,   20,    8,    1;\n  4:   32,   56,   36,   10,    1;\n  5:   64,  144,  128,   56,   12,    1;\n  6:  128,  352,  400,  240,   80,   14,   1;\n  7:  256,  832, 1152,  880,  400,  108,  16,   1;\n  8:  512, 1920, 3136, 2912, 1680,  616, 140,  18,  1;\n  9: 1024, 4352, 8192, 8960, 6272, 2912, 896, 176, 20, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1])/(n - 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-92128fbba2409faa","steps":["basis A007318"],"edges":["e-bc413433eedeab55"],"text":"A093560 \u2192 basis A007318"},"headline":"A093560 \u2192 basis A007318","chain":null}