{"id":"n-82aad7cb8d2321e5","revision":0,"first_index":1,"fingerprint":"f6-b821bbf6b18e27b2","name":"basis_map_A013609 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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["2","2"],["6","10","4"],["18","42","32","8"],["54","162","180","88","16"],["162","594","864","624","224","32"],["486","2106","3780","3600","1920","544","64"],["1458","7290","15552","18360","12960","5472","1280","128"],["4374","24786","61236","86184","75600","42336","14784","2944","256"],["13122","83106","233280","381024","399168","278208","129024","38400","6656","512"],["39366","275562","866052","1609632","1959552","1632960","943488","373248","96768","14848","1024"],["118098","905418","3149280","6561000","9097920","8817984","6096384","3006720","1036800","238080","32768","2048"],["354294","2952450","11258676","25981560","40415760","44649792","35925120","21212928","9123840","2787840","574464","71680","4096"],["1062882","9565938","39680928","100462032","173210400","214780896","197074944","135489024","69797376","26611200","7299072","1363968","155648","8192"],["3188646","30823578","138174660","380747952","720555264","990763488","1020786624","800616960","480370176","219428352","75119616","18690048","3194880","335872","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","20","100","500","2500","12500","62500","312500","1562500","7812500","39062500","195312500","976562500","4882812500"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-4"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199006,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-3a05dfddaf9200b8"]},{"id":199007,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-3a05dfddaf9200b8"]},{"id":199008,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-3a05dfddaf9200b8"]},{"id":199009,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-3a05dfddaf9200b8"]}],"region_exact":{},"incoming_edges":[{"id":"e-3a05dfddaf9200b8","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-82aad7cb8d2321e5","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":"a4d5421a98b4ab485c824b52db3845d2771fbf9ae279237bfb8b92ac6a9958db","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f2a2a3d7ea29bcbf","role":"sequence","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.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-01314641cd86215f","kind":"fingerprint_related","node_a_id":"n-82aad7cb8d2321e5","node_b_id":"n-079690ff502c56c6","fingerprint":"f6-b821bbf6b18e27b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b821bbf6b18e27b2"},"other_id":"n-079690ff502c56c6","other":{"id":"n-079690ff502c56c6","name":"basis_map_A013609 of Triangular matrix arising in enumeration of catafusenes, read by rows.","first_index":0,"fingerprint":"f6-b821bbf6b18e27b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","20","12"]],"card_rows":[["1"],["2","2"],["8","20","12"],["32","128","168","72"]],"row_sums_preview":["1","4","40","400","4000","40000","400000","4000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-685ab5842fbad7f4","steps":["basis A013609"],"edges":["e-a7392d08bc47ef7e"],"text":"A038763 \u2192 basis A013609"},"headline":"A038763 \u2192 basis A013609","chain":null}},{"id":"l-0ffb8db423e173b8","kind":"fingerprint_related","node_a_id":"n-82aad7cb8d2321e5","node_b_id":"n-fd95b73866112a83","fingerprint":"f6-b821bbf6b18e27b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b821bbf6b18e27b2"},"other_id":"n-fd95b73866112a83","other":{"id":"n-fd95b73866112a83","name":"basis_map_A013610 of A127899(unsigned) * A007318.","first_index":0,"fingerprint":"f6-b821bbf6b18e27b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","6"],["18","45","27"]],"card_rows":[["1"],["6","6"],["18","45","27"],["48","192","252","108"]],"row_sums_preview":["1","12","90","600","3750","22500","131250","750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e917c954825de4ec","steps":["basis A013610"],"edges":["e-1326230c683b860b"],"text":"A134239 \u2192 basis A013610"},"headline":"A134239 \u2192 basis A013610","chain":null}},{"id":"l-5a926df0b12d7921","kind":"fingerprint_related","node_a_id":"n-82aad7cb8d2321e5","node_b_id":"n-b799b709bd4d1eca","fingerprint":"f6-b821bbf6b18e27b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b821bbf6b18e27b2"},"other_id":"n-b799b709bd4d1eca","other":{"id":"n-b799b709bd4d1eca","name":"basis_map_A013610 of Numbers in (2,1)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-b821bbf6b18e27b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["6","15","9"]],"card_rows":[["1"],["3","3"],["6","15","9"],["12","48","63","27"]],"row_sums_preview":["1","6","30","150","750","3750","18750","93750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c5f5be79d9d1ae3","steps":["basis A013610"],"edges":["e-cde763b079e4126c"],"text":"A029653 \u2192 basis A013610"},"headline":"A029653 \u2192 basis A013610","chain":null}},{"id":"l-eaa303e5a4a9c2d1","kind":"fingerprint_related","node_a_id":"n-1ce1c7eadcd8d479","node_b_id":"n-82aad7cb8d2321e5","fingerprint":"f6-b821bbf6b18e27b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b821bbf6b18e27b2"},"other_id":"n-1ce1c7eadcd8d479","other":{"id":"n-1ce1c7eadcd8d479","name":"obreschkoff_combination of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-b821bbf6b18e27b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","3"],["6","15","9"],["12","48","63","27"]],"card_rows":[["3","3"],["6","15","9"],["12","48","63","27"],["24","132","270","243","81"]],"row_sums_preview":["6","30","150","750","3750","18750","93750","468750"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-cfb9e27fdcb193b1"],"text":"A013620 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A013620 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,     2;\n   3:     6,    10,      4;\n   4:    18,    42,     32,      8;\n   5:    54,   162,    180,     88,     16;\n   6:   162,   594,    864,    624,    224,     32;\n   7:   486,  2106,   3780,   3600,   1920,    544,     64;\n   8:  1458,  7290,  15552,  18360,  12960,   5472,   1280,   128;\n   9:  4374, 24786,  61236,  86184,  75600,  42336,  14784,  2944,  256;\n  10: 13122, 83106, 233280, 381024, 399168, 278208, 129024, 38400, 6656, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((3*n - 6 + (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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":["basis A013609"],"edges":["e-3a05dfddaf9200b8"],"text":"A200139 \u2192 basis A013609"},"headline":"A200139 \u2192 basis A013609","chain":null}