{"id":"n-af7f3f008db46a2b","revision":0,"first_index":0,"fingerprint":"f6-04668acf2e74f66d","name":"basis_map_A013609 of Triangle A127647 * A097806, read by rows.","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"],["2","2"],["4","12","8"],["6","30","48","24"],["10","70","180","200","80"],["16","144","512","896","768","256"],["26","286","1300","3120","4160","2912","832"],["42","546","3024","9240","16800","18144","10752","2688"],["68","1020","6664","24752","57120","83776","76160","39168","8704"],["110","1870","14080","61600","172480","320320","394240","309760","140800","28160"],["178","3382","28836","145248","478464","1076544","1674624","1777152","1230336","501248","91136"],["288","6048","57600","328320","1244160","3290112","6193152","8294400","7741440","4792320","1769472","294912"],["466","10718","112772","717640","3075600","9349824","20668032","33462528","39367680","32806400","18371584","6203392","954368"],["754","18850","217152","1526096","7298720","25081056","63697920","121026048","171984384","180477440","135888896","69488640","21618688","3088384"],["1220","32940","412360","3172000","16748160","64201280","184229760","401955840","669926400","848573440","803911680","552181760","259850240","74956800","9994240"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","24","108","540","2592","12636","61236","297432","1443420","7007148","34012224","165101004","801413028","3890148120"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["4","12","8"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8","-8"],["1","4","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-4"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198589,"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-6bc1e4a45d686af0"]},{"id":198590,"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-6bc1e4a45d686af0"]},{"id":198591,"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-6bc1e4a45d686af0"]}],"region_exact":{},"incoming_edges":[{"id":"e-6bc1e4a45d686af0","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-af7f3f008db46a2b","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":"fbc929e6eb9d26b2f2eef2b9b5392f8c0581c0b663c89ff6a661a61f5b295416","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-a009f9b98df3ca1d","role":"sequence","name":"Triangle A127647 * A097806, read by rows.","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-2f1a56d0f771b285","kind":"fingerprint_related","node_a_id":"n-b427d13cbd1f8c56","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-b427d13cbd1f8c56","other":{"id":"n-b427d13cbd1f8c56","name":"basis_map_A007318 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":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","12","4"]],"card_rows":[["1"],["2","2"],["8","12","4"],["32","64","40","8"]],"row_sums_preview":["1","4","24","144","864","5184","31104","186624"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-19598058c018a2ec","steps":["basis A013609"],"edges":["e-09ee5a717c6cfde3"],"text":"A136158 \u2192 basis A013609"},"headline":"A136158 \u2192 basis A013609","chain":null}},{"id":"l-3e7138d78fe17370","kind":"fingerprint_related","node_a_id":"n-dd72b1be3ab7cf4c","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-dd72b1be3ab7cf4c","other":{"id":"n-dd72b1be3ab7cf4c","name":"basis_map_A013609 of Riordan array (1+x, 1) read by rows.","first_index":0,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["2","6","4"]],"card_rows":[["1"],["2","2"],["2","6","4"],["2","10","16","8"]],"row_sums_preview":["1","4","12","36","108","324","972","2916"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-38c9f89c1db150a9","steps":["basis A013609"],"edges":["e-00b5b4cc54f2d3a6"],"text":"A097806 \u2192 basis A013609"},"headline":"A097806 \u2192 basis A013609","chain":null}},{"id":"l-411df6591e44ea71","kind":"fingerprint_related","node_a_id":"n-b8a487699b52b31f","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-b8a487699b52b31f","other":{"id":"n-b8a487699b52b31f","name":"obreschkoff_combination of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2","2"],["2","6","4"],["2","10","16","8"]],"card_rows":[["2","2"],["2","6","4"],["2","10","16","8"],["2","14","36","40","16"]],"row_sums_preview":["4","12","36","108","324","972","2916","8748"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-1774203d413cf9ca"],"text":"A013609 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A013609 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}},{"id":"l-51cc6a36bc7780da","kind":"fingerprint_related","node_a_id":"n-e662b9c4c7bb9dd2","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-e662b9c4c7bb9dd2","other":{"id":"n-e662b9c4c7bb9dd2","name":"basis_map_A007318 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":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","3","1"]],"card_rows":[["1"],["1","1"],["2","3","1"],["4","8","5","1"]],"row_sums_preview":["1","2","6","18","54","162","486","1458"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3783f6df357ea675","steps":["basis A007318"],"edges":["e-d87f7641f785a341"],"text":"A097805 \u2192 basis A007318"},"headline":"A097805 \u2192 basis A007318","chain":null}},{"id":"l-551e68ea69ba6448","kind":"fingerprint_related","node_a_id":"n-f2a2a3d7ea29bcbf","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-f2a2a3d7ea29bcbf","other":{"id":"n-f2a2a3d7ea29bcbf","name":"Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","first_index":1,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","3","1"]],"card_rows":[["1"],["1","1"],["2","3","1"],["4","8","5","1"]],"row_sums_preview":["1","2","6","18","54","162","486","1458"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A200139"],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":[],"edges":[],"text":"A200139"},"headline":"A200139: Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","chain":null}},{"id":"l-72d9a3beff0055da","kind":"fingerprint_related","node_a_id":"n-af7f3f008db46a2b","node_b_id":"n-550f42ea21954e22","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-550f42ea21954e22","other":{"id":"n-550f42ea21954e22","name":"basis_map_A013611 of (3,1) Pascal triangle.","first_index":0,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["8","24","16"]],"card_rows":[["1"],["4","4"],["8","24","16"],["16","80","128","64"]],"row_sums_preview":["1","8","48","288","1728","10368","62208","373248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-92128fbba2409faa","steps":["basis A013611"],"edges":["e-342cfe46a0a2cd4c"],"text":"A093560 \u2192 basis A013611"},"headline":"A093560 \u2192 basis A013611","chain":null}},{"id":"l-92a1be590dbf37fa","kind":"fingerprint_related","node_a_id":"n-29ea4bcda6322811","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-29ea4bcda6322811","other":{"id":"n-29ea4bcda6322811","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["6","9","3"]],"card_rows":[["1"],["2","2"],["6","9","3"],["16","32","20","4"]],"row_sums_preview":["1","4","18","72","270","972","3402","11664"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a88a7983f557eac4","steps":["basis A007318"],"edges":["e-39c185df7b68a75c"],"text":"A127952 \u2192 basis A007318"},"headline":"A127952 \u2192 basis A007318","chain":null}},{"id":"l-c6c72b73198aa4db","kind":"fingerprint_related","node_a_id":"n-4a90276c6d0931df","node_b_id":"n-af7f3f008db46a2b","fingerprint":"f6-04668acf2e74f66d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-04668acf2e74f66d"},"other_id":"n-4a90276c6d0931df","other":{"id":"n-4a90276c6d0931df","name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-04668acf2e74f66d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["6","6"],["6","18","12"],["6","30","48","24"]],"card_rows":[["6","6"],["6","18","12"],["6","30","48","24"],["6","42","108","120","48"]],"row_sums_preview":["12","36","108","324","972","2916","8748","26244"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-6efef9a3e4d28a62"],"text":"A120909 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A120909 \u2192 P\u2099 + P\u2099\u208a\u2081","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:   4,   12,     8;\n  3:   6,   30,    48,    24;\n  4:  10,   70,   180,   200,     80;\n  5:  16,  144,   512,   896,    768,    256;\n  6:  26,  286,  1300,  3120,   4160,   2912,    832;\n  7:  42,  546,  3024,  9240,  16800,  18144,  10752,   2688;\n  8:  68, 1020,  6664, 24752,  57120,  83776,  76160,  39168,   8704;\n  9: 110, 1870, 14080, 61600, 172480, 320320, 394240, 309760, 140800, 28160;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-2+(4*n-8)*x+(4*n-8)*x^2)*P(n-2,x) + (n-2+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 4 + 12*x + 8*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 4 + 12*x + 8*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*P[n - 2, x] + (n - 2 + (2*n - 4)*x)*P[n - 1, 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] = 4 + 12*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 8)*x + (4*n - 8)*x^2)*v[i - 2] + (n - 2 + (2*n - 4)*x)*v[i - 1])/(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-a009f9b98df3ca1d","steps":["basis A013609"],"edges":["e-6bc1e4a45d686af0"],"text":"A128540 \u2192 basis A013609"},"headline":"A128540 \u2192 basis A013609","chain":null}