{"id":"n-3a1fea9eacf93d7c","revision":0,"first_index":0,"fingerprint":"f6-e3ba37fff5d43b63","name":"basis_map_A013611 of basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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"],["3","4"],["15","32","16"],["105","284","240","64"],["945","2976","3296","1536","256"],["10395","36516","48160","30080","8960","1024"],["135135","516288","772144","583680","236800","49152","4096"],["2027025","8284860","13647312","11843776","5886720","1684480","258048","16384"],["34459425","148950720","265143744","255933440","147449344","52183040","11124736","1310720","65536"],["654729075","2967901380","5633534016","5923310336","3825271296","1581275136","420102144","69402624","6488064","262144"],["13749310575","64944845280","130175819856","146923653120","104023938560","48507863040","15147245568","3137863680","413859840","31457280","1048576"],["316234143225","1548728683740","3253823237808","3899947301184","2980245199360","1531776604160","542418100224","132759846912","22070231040","2378956800","149946368","4194304"],["7905853580625","39983153666400","87540495680160","110513975480832","90105919188736","50215395901440","19687558922240","5488668573696","1082795163648","147754844160","13264486400","704643072","16777216"],["213458046676875","1111168563315300","2523525998029920","3334039320703104","2874915720019200","1716239366093824","732425674506240","226944287178752","51190143713280","8320561446912","949160509440","72083308544","3271557120","67108864"],["6190283353629375","33077720522851200","77626928196128880","106781244292509696","96708713163369216","61270604496797696","28105302025056256","9511087026208768","2392291316400128","446056856813568","60807900561408","5887057985536","383208390656","15032385536","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","63","693","9009","135135","2297295","43648605","916620705","21082276215","527056905375","14230536445125","412685556908625","12793252264167375","422177324717523375"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205945,"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-b092cd8eb676385f"]},{"id":205946,"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-b092cd8eb676385f"]},{"id":205947,"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-b092cd8eb676385f"]},{"id":205948,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-b092cd8eb676385f"]}],"region_exact":{},"incoming_edges":[{"id":"e-b092cd8eb676385f","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-3a1fea9eacf93d7c","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":"ceacb8d72e96a1ac51802f425480170e385ac8c1a93e70afb6cb608dcad596bd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fdb0bda32eba4618","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","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-1b17232ba6bf4d1b","kind":"fingerprint_related","node_a_id":"n-1b004f3b09ec251c","node_b_id":"n-3a1fea9eacf93d7c","fingerprint":"f6-e3ba37fff5d43b63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e3ba37fff5d43b63"},"other_id":"n-1b004f3b09ec251c","other":{"id":"n-1b004f3b09ec251c","name":"reversal of basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-e3ba37fff5d43b63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","3"],["16","32","15"]],"card_rows":[["1"],["4","3"],["16","32","15"],["64","240","284","105"]],"row_sums_preview":["1","7","63","693","9009","135135","2297295","43648605"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013609","reverse"],"edges":["e-5c9ad73da84eb2a7","e-77797de001f334d4"],"text":"A161198 \u2192 basis A013609 \u2192 reverse"},"headline":"A161198 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-2143b7b464a16e68","kind":"fingerprint_related","node_a_id":"n-3492b095602a362c","node_b_id":"n-3a1fea9eacf93d7c","fingerprint":"f6-e3ba37fff5d43b63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e3ba37fff5d43b63"},"other_id":"n-3492b095602a362c","other":{"id":"n-3492b095602a362c","name":"basis_map_A013609 of basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-e3ba37fff5d43b63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","4"],["45","96","48"]],"card_rows":[["1"],["3","4"],["45","96","48"],["1575","4260","3600","960"]],"row_sums_preview":["1","7","189","10395","945945","127702575","23880381525","5898454236675"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013609","basis A013609"],"edges":["e-2613ba1fbc2604f1","e-b8fb90213ff71ad1"],"text":"A204420 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A204420 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-64590c8dcecd4b95","kind":"fingerprint_related","node_a_id":"n-67a3afb389f8b05c","node_b_id":"n-3a1fea9eacf93d7c","fingerprint":"f6-e3ba37fff5d43b63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e3ba37fff5d43b63"},"other_id":"n-67a3afb389f8b05c","other":{"id":"n-67a3afb389f8b05c","name":"basis_map_A013609 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","first_index":1,"fingerprint":"f6-e3ba37fff5d43b63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","4"],["15","32","16"]],"card_rows":[["1"],["3","4"],["15","32","16"],["105","284","240","64"]],"row_sums_preview":["1","7","63","693","9009","135135","2297295","43648605"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013609"],"edges":["e-5c9ad73da84eb2a7"],"text":"A161198 \u2192 basis A013609"},"headline":"A161198 \u2192 basis A013609","chain":null}},{"id":"l-dddddb31cfb9121c","kind":"fingerprint_related","node_a_id":"n-8a2660088131c4ff","node_b_id":"n-3a1fea9eacf93d7c","fingerprint":"f6-e3ba37fff5d43b63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e3ba37fff5d43b63"},"other_id":"n-8a2660088131c4ff","other":{"id":"n-8a2660088131c4ff","name":"basis_map_A013609 of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","first_index":0,"fingerprint":"f6-e3ba37fff5d43b63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","8"],["60","128","64"]],"card_rows":[["1"],["6","8"],["60","128","64"],["840","2272","1920","512"]],"row_sums_preview":["1","14","252","5544","144144","4324320","147026880","5587021440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013611","basis A013609"],"edges":["e-403803ced55fc794","e-2af578e9e8b1a2fe"],"text":"A290319 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A290319 \u2192 basis A013611 \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:         1;\n  1:         3,          4;\n  2:        15,         32,         16;\n  3:       105,        284,        240,         64;\n  4:       945,       2976,       3296,       1536,        256;\n  5:     10395,      36516,      48160,      30080,       8960,       1024;\n  6:    135135,     516288,     772144,     583680,     236800,      49152,      4096;\n  7:   2027025,    8284860,   13647312,   11843776,    5886720,    1684480,    258048,    16384;\n  8:  34459425,  148950720,  265143744,  255933440,  147449344,   52183040,  11124736,  1310720,   65536;\n  9: 654729075, 2967901380, 5633534016, 5923310336, 3825271296, 1581275136, 420102144, 69402624, 6488064, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n + 1 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + 1 + 4*x)*v[i - 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-4f6e747b794bcc2e","steps":["x \u21a6 x+1","basis A013611"],"edges":["e-29cb0c45f74a7f15","e-b092cd8eb676385f"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 basis A013611"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 basis A013611","chain":null}