{"id":"n-3a64095a81c262d2","revision":0,"first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","name":"basis_map_A013609 of Pascal (1,3) 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":[["3"],["4","6"],["8","20","12"],["16","56","64","24"],["32","144","240","176","48"],["64","352","768","832","448","96"],["128","832","2240","3200","2560","1088","192"],["256","1920","6144","10880","11520","7296","2560","384"],["512","4352","16128","34048","44800","37632","19712","5888","768"],["1024","9728","40960","100352","157696","164864","114688","51200","13312","1536"],["2048","21504","101376","282624","516096","645120","559104","331776","129024","29696","3072"],["4096","47104","245760","768000","1597440","2322432","2408448","1781760","921600","317440","65536","6144"],["8192","102400","585728","2027520","4730880","7839744","9461760","8380416","5406720","2478080","765952","143360","12288"],["16384","221184","1376256","5226496","13516800","25141248","34603008","35684352","27574272","15769600","6488064","1818624","311296","24576"],["32768","475136","3194880","13205504","37486592","77316096","119488512","140574720","126517248","86687744","44515328","16613376","4259840","671744","49152"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","10","40","160","640","2560","10240","40960","163840","655360","2621440","10485760","41943040","167772160","671088640"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"],["4","6"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-2"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":198739,"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-93be59244d138bf4"]},{"id":198740,"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-93be59244d138bf4"]},{"id":198741,"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-93be59244d138bf4"]},{"id":198742,"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-93be59244d138bf4"]}],"region_exact":{},"incoming_edges":[{"id":"e-93be59244d138bf4","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3a64095a81c262d2","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":"be6d84f3f7719344880403cfaeb96232209071c69a045672e638c6c0b63e640c","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-c5cd4d1a5bdd7e74","role":"sequence","name":"Pascal (1,3) 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-055dc23fce57684f","kind":"fingerprint_related","node_a_id":"n-265c490839a08efd","node_b_id":"n-3a64095a81c262d2","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-265c490839a08efd","other":{"id":"n-265c490839a08efd","name":"Numbers in (3,2)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["3","5","2"]],"card_rows":[["1"],["3","2"],["3","5","2"],["3","8","7","2"]],"row_sums_preview":["1","5","10","20","40","80","160","320"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A029618"],"derivation":{"root":"n-265c490839a08efd","steps":[],"edges":[],"text":"A029618"},"headline":"A029618: Numbers in (3,2)-Pascal triangle (by row).","chain":null}},{"id":"l-20d02e27f34518ec","kind":"fingerprint_related","node_a_id":"n-3a64095a81c262d2","node_b_id":"n-3315275f16fbe7b5","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-3315275f16fbe7b5","other":{"id":"n-3315275f16fbe7b5","name":"basis_map_A013610 of 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":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["6","15","9"]],"card_rows":[["1"],["2","3"],["6","15","9"],["18","63","72","27"]],"row_sums_preview":["1","5","30","180","1080","6480","38880","233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":["basis A013610"],"edges":["e-259befa5e0ac300b"],"text":"A200139 \u2192 basis A013610"},"headline":"A200139 \u2192 basis A013610","chain":null}},{"id":"l-2e1734db3c80aa4c","kind":"fingerprint_related","node_a_id":"n-aa480b4ac960c965","node_b_id":"n-3a64095a81c262d2","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-aa480b4ac960c965","other":{"id":"n-aa480b4ac960c965","name":"Numbers in the (2,3)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["2","5","3"]],"card_rows":[["1"],["2","3"],["2","5","3"],["2","7","8","3"]],"row_sums_preview":["1","5","10","20","40","80","160","320"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A029600"],"derivation":{"root":"n-aa480b4ac960c965","steps":[],"edges":[],"text":"A029600"},"headline":"A029600: Numbers in the (2,3)-Pascal triangle (by row).","chain":null}},{"id":"l-3288484fce573d34","kind":"fingerprint_related","node_a_id":"n-77cf2a9e7dd2c35b","node_b_id":"n-3a64095a81c262d2","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-77cf2a9e7dd2c35b","other":{"id":"n-77cf2a9e7dd2c35b","name":"basis_map_A013609 of Numbers in (2,1)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["6","10","4"]],"card_rows":[["1"],["3","2"],["6","10","4"],["12","32","28","8"]],"row_sums_preview":["1","5","20","80","320","1280","5120","20480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c5f5be79d9d1ae3","steps":["basis A013609"],"edges":["e-446c00036ed40438"],"text":"A029653 \u2192 basis A013609"},"headline":"A029653 \u2192 basis A013609","chain":null}},{"id":"l-3ed5a1e615cd3452","kind":"fingerprint_related","node_a_id":"n-2224fadf43dd48ee","node_b_id":"n-3a64095a81c262d2","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-2224fadf43dd48ee","other":{"id":"n-2224fadf43dd48ee","name":"derivative of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","first_index":1,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["0","2","3"],["0","0","0","4","10","6"]],"card_rows":[["0","2","3"],["0","0","0","4","10","6"],["0","0","0","0","0","6","21","24","9"],["0","0","0","0","0","0","0","8","36","60","44","12"]],"row_sums_preview":["0","5","20","60","160","400","960","2240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-11b174a1f5e78a68","steps":["D"],"edges":["e-e26a694cd197a81e"],"text":"Veronese parent of A098172 \u2192 D"},"headline":"Veronese parent of A098172 \u2192 D","chain":null}},{"id":"l-a78afb4ce0ef2683","kind":"fingerprint_related","node_a_id":"n-4b8d8a8d07abc4f3","node_b_id":"n-3a64095a81c262d2","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-4b8d8a8d07abc4f3","other":{"id":"n-4b8d8a8d07abc4f3","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","2"],["6","10","4"],["12","32","28","8"]],"card_rows":[["3","2"],["6","10","4"],["12","32","28","8"],["24","88","120","72","16"]],"row_sums_preview":["5","20","80","320","1280","5120","20480","81920"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-ec9ffad953fe7bfb"],"text":"A038208 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038208 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}},{"id":"l-b895a2ac09404aca","kind":"fingerprint_related","node_a_id":"n-3a64095a81c262d2","node_b_id":"n-ed2a76556fde8ee6","fingerprint":"f6-8a8f26bfa265b5f7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8a8f26bfa265b5f7"},"other_id":"n-ed2a76556fde8ee6","other":{"id":"n-ed2a76556fde8ee6","name":"basis_map_A013609 of A127899(unsigned) * A007318.","first_index":0,"fingerprint":"f6-8a8f26bfa265b5f7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","4"],["18","30","12"]],"card_rows":[["1"],["6","4"],["18","30","12"],["48","128","112","32"]],"row_sums_preview":["1","10","60","320","1600","7680","35840","163840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e917c954825de4ec","steps":["basis A013609"],"edges":["e-2086690850cdd78c"],"text":"A134239 \u2192 basis A013609"},"headline":"A134239 \u2192 basis A013609","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:    3;\n  1:    4,    6;\n  2:    8,   20,    12;\n  3:   16,   56,    64,     24;\n  4:   32,  144,   240,    176,     48;\n  5:   64,  352,   768,    832,    448,     96;\n  6:  128,  832,  2240,   3200,   2560,   1088,    192;\n  7:  256, 1920,  6144,  10880,  11520,   7296,   2560,   384;\n  8:  512, 4352, 16128,  34048,  44800,  37632,  19712,  5888,   768;\n  9: 1024, 9728, 40960, 100352, 157696, 164864, 114688, 51200, 13312, 1536;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(2*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (2*n - 2)*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] = 3; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (2*n - 2)*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-c5cd4d1a5bdd7e74","steps":["basis A013609"],"edges":["e-93be59244d138bf4"],"text":"A095660 \u2192 basis A013609"},"headline":"A095660 \u2192 basis A013609","chain":null}
