{"id":"n-079690ff502c56c6","revision":0,"first_index":0,"fingerprint":"f6-b821bbf6b18e27b2","name":"basis_map_A013609 of Triangular matrix arising in enumeration of catafusenes, 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"],["8","20","12"],["32","128","168","72"],["128","704","1440","1296","432"],["512","3584","9984","13824","9504","2592"],["2048","17408","61440","115200","120960","67392","15552"],["8192","81920","350208","829440","1175040","995328","466560","93312"],["32768","376832","1892352","5419008","9676800","11031552","7838208","3172608","559872"],["131072","1703936","9830400","33030144","71221248","102187008","97542144","59719680","21275136","3359232"],["524288","7602176","49545216","191102976","483065856","836075520","1003290624","824131584","443418624","141087744","20155392"],["2097152","33554432","243793920","1061683200","3078881280","6242697216","9029615616","9316270080","6718464000","3224862720","927148032","120932352"],["8388608","146800640","1176502272","5709496320","18685624320","43444076544","73574645760","91442774016","82771476480","53210234880","23057768448","6046617600","725594112"],["33554432","637534208","5586812928","29896998912","108999475200","285890052096","554963042304","807218970624","879742550016","709469798400","411492483072","162533081088","39182082048","4353564672"],["134217728","2751463424","26172456960","153108873216","615379894272","1797557059584","3935192481792","6558654136320","8362284023808","8116334493696","5902788722688","3119087222784","1131926814720","252506750976","26121388032"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","40","400","4000","40000","400000","4000000","40000000","400000000","4000000000","40000000000","400000000000","4000000000000","40000000000000"],"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"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-6"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199239,"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-a7392d08bc47ef7e"]},{"id":199240,"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-a7392d08bc47ef7e"]},{"id":199241,"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-a7392d08bc47ef7e"]},{"id":199242,"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-a7392d08bc47ef7e"]}],"region_exact":{},"incoming_edges":[{"id":"e-a7392d08bc47ef7e","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-079690ff502c56c6","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":"5f3e38d4cceabcbcdc5b13ca20d6651cbce0d6eecc4464dd811fab81dc74fcdb","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-685ab5842fbad7f4","role":"sequence","name":"Triangular matrix arising in enumeration of catafusenes, 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-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-82aad7cb8d2321e5","other":{"id":"n-82aad7cb8d2321e5","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.","first_index":1,"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"],["6","10","4"]],"card_rows":[["1"],["2","2"],["6","10","4"],["18","42","32","8"]],"row_sums_preview":["1","4","20","100","500","2500","12500","62500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":["basis A013609"],"edges":["e-3a05dfddaf9200b8"],"text":"A200139 \u2192 basis A013609"},"headline":"A200139 \u2192 basis A013609","chain":null}},{"id":"l-6df46876ab7e12da","kind":"fingerprint_related","node_a_id":"n-079690ff502c56c6","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-80c9226942fe2ff4","kind":"fingerprint_related","node_a_id":"n-1ce1c7eadcd8d479","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-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}},{"id":"l-a9153307412e840f","kind":"fingerprint_related","node_a_id":"n-079690ff502c56c6","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}}],"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:      8,      20,      12;\n  3:     32,     128,     168,       72;\n  4:    128,     704,    1440,     1296,      432;\n  5:    512,    3584,    9984,    13824,     9504,      2592;\n  6:   2048,   17408,   61440,   115200,   120960,     67392,    15552;\n  7:   8192,   81920,  350208,   829440,  1175040,    995328,   466560,    93312;\n  8:  32768,  376832, 1892352,  5419008,  9676800,  11031552,  7838208,  3172608,   559872;\n  9: 131072, 1703936, 9830400, 33030144, 71221248, 102187008, 97542144, 59719680, 21275136, 3359232;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(6*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (6*n - 6)*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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (6*n - 6)*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-685ab5842fbad7f4","steps":["basis A013609"],"edges":["e-a7392d08bc47ef7e"],"text":"A038763 \u2192 basis A013609"},"headline":"A038763 \u2192 basis A013609","chain":null}
