{"id":"n-c1326eae95374f9d","revision":0,"first_index":0,"fingerprint":"f6-432dc7cc45625337","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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"],["4","11"],["16","88","121"],["64","528","1452","1331"],["256","2816","11616","21296","14641"],["1024","14080","77440","212960","292820","161051"],["4096","67584","464640","1703680","3513840","3865224","1771561"],["16384","315392","2601984","11925760","32795840","54113136","49603708","19487171"],["65536","1441792","13877248","76324864","262366720","577206784","793659328","623589472","214358881"],["262144","6488064","71368704","457949184","1889040384","5194861056","9523911936","11224610496","7716919716","2357947691"],["1048576","28835840","356843520","2616852480","12593602560","41558888448","95239119360","149661473280","154338394320","94317907640","25937424601"],["4194304","126877696","1744568320","14392688640","79159787520","304765181952","838104250368","1646276206080","2263629783360","2074993968080","1141246682444","285311670611"],["16777216","553648128","8373927936","76761006080","474958725120","2089818390528","6704834002944","15804251578368","27163557400320","33199903489280","27389920378656","13694960189328","3138428376721"],["67108864","2399141888","39585841152","399157231616","2744205967360","13583819538432","49807338307584","136970180345856","282500996963328","431598745360640","474758619896704","356068964922528","163198275589492","34522712143931"],["268435456","10334765056","184733925376","2032073179136","15367553417216","84521543794688","348651368153088","1095761442766848","2636675971657728","4833905948039168","6646620678553856","6646620678553856","4569551716505776","1933271880060136","379749833583241"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","225","3375","50625","759375","11390625","170859375","2562890625","38443359375","576650390625","8649755859375","129746337890625","1946195068359375","29192926025390625"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188788,"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-13b8d53c469bc767"]},{"id":188789,"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-13b8d53c469bc767"]},{"id":188790,"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-13b8d53c469bc767"]}],"region_exact":{},"incoming_edges":[{"id":"e-13b8d53c469bc767","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c1326eae95374f9d","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"427407889f3556dad31f0f9836995e25c894ed757f9c72bb25963f1ecf2b76b9","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-e03395d80bc5749d","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","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-4d7243a9ba95e7b3","kind":"fingerprint_related","node_a_id":"n-c1326eae95374f9d","node_b_id":"n-f2db2a8b2f7dd451","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-f2db2a8b2f7dd451","other":{"id":"n-f2db2a8b2f7dd451","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["792","2178"],["14256","78408","107811"],["228096","1881792","5174928","4743684"]],"card_rows":[["792","2178"],["14256","78408","107811"],["228096","1881792","5174928","4743684"],["3421440","37635840","155247840","284621040","195676965"]],"row_sums_preview":["2970","200475","12028500","676603125","36536568750","1918169859375","98648735625000","4994092241015625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013610","D"],"edges":["e-4217afbeac9f09d9","e-b3f6922769133f5e"],"text":"A013618 \u2192 basis A013610 \u2192 D"},"headline":"A013618 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-5305179d5a493ccc","kind":"fingerprint_related","node_a_id":"n-c1326eae95374f9d","node_b_id":"n-b1d961b0981a7a0a","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-b1d961b0981a7a0a","other":{"id":"n-b1d961b0981a7a0a","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","352","64"],["10648","11616","4224","512"],["234256","340736","185856","45056","4096"]],"card_rows":[["484","352","64"],["10648","11616","4224","512"],["234256","340736","185856","45056","4096"],["5153632","9370240","6814720","2478080","450560","32768"]],"row_sums_preview":["900","27000","810000","24300000","729000000","21870000000","656100000000","19683000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611","basis A007318"],"edges":["e-0eb3975e7d8d74fc","e-2ca04c1400b7e888"],"text":"A038328 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038328 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-575391712be4a7ff","kind":"fingerprint_related","node_a_id":"n-c1326eae95374f9d","node_b_id":"n-456ae85670c94210","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-456ae85670c94210","other":{"id":"n-456ae85670c94210","name":"reversal of basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1089","792","144"],["35937","39204","14256","1728"],["1185921","1724976","940896","228096","20736"]],"card_rows":[["1089","792","144"],["35937","39204","14256","1728"],["1185921","1724976","940896","228096","20736"],["39135393","71155260","51749280","18817920","3421440","248832"]],"row_sums_preview":["2025","91125","4100625","184528125","8303765625","373669453125","16815125390625","756680642578125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013610","reverse"],"edges":["e-4217afbeac9f09d9","e-1446027a58c5a6c0"],"text":"A013618 \u2192 basis A013610 \u2192 reverse"},"headline":"A013618 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-8e1c8c2de6e52b3d","kind":"fingerprint_related","node_a_id":"n-ea3be122f7c7526c","node_b_id":"n-c1326eae95374f9d","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-ea3be122f7c7526c","other":{"id":"n-ea3be122f7c7526c","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","792","1089"],["1728","14256","39204","35937"],["20736","228096","940896","1724976","1185921"]],"card_rows":[["144","792","1089"],["1728","14256","39204","35937"],["20736","228096","940896","1724976","1185921"],["248832","3421440","18817920","51749280","71155260","39135393"]],"row_sums_preview":["2025","91125","4100625","184528125","8303765625","373669453125","16815125390625","756680642578125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013610"],"edges":["e-4217afbeac9f09d9"],"text":"A013618 \u2192 basis A013610"},"headline":"A013618 \u2192 basis A013610","chain":null}},{"id":"l-b2d01fd42a83b59a","kind":"fingerprint_related","node_a_id":"n-f74c73498eaf58f5","node_b_id":"n-c1326eae95374f9d","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-f74c73498eaf58f5","other":{"id":"n-f74c73498eaf58f5","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["88","32"]],"card_rows":[["4"],["88","32"],["1452","1056","192"],["21296","23232","8448","1024"]],"row_sums_preview":["0","4","120","2700","54000","1012500","18225000","318937500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013611","D"],"edges":["e-551d764375152d3c","e-3f8d12cd2772fcb6"],"text":"A038303 \u2192 basis A013611 \u2192 D"},"headline":"A038303 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-dfb1be84483c422c","kind":"fingerprint_related","node_a_id":"n-e03395d80bc5749d","node_b_id":"n-c1326eae95374f9d","fingerprint":"f6-432dc7cc45625337","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-432dc7cc45625337"},"other_id":"n-e03395d80bc5749d","other":{"id":"n-e03395d80bc5749d","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-432dc7cc45625337","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","4"],["121","88","16"]],"card_rows":[["1"],["11","4"],["121","88","16"],["1331","1452","528","64"]],"row_sums_preview":["1","15","225","3375","50625","759375","11390625","170859375"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013611"],"edges":["e-551d764375152d3c"],"text":"A038303 \u2192 basis A013611"},"headline":"A038303 \u2192 basis A013611","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:      4,      11;\n  2:     16,      88,      121;\n  3:     64,     528,     1452,      1331;\n  4:    256,    2816,    11616,     21296,      14641;\n  5:   1024,   14080,    77440,    212960,     292820,     161051;\n  6:   4096,   67584,   464640,   1703680,    3513840,    3865224,    1771561;\n  7:  16384,  315392,  2601984,  11925760,   32795840,   54113136,   49603708,    19487171;\n  8:  65536, 1441792, 13877248,  76324864,  262366720,  577206784,  793659328,   623589472,  214358881;\n  9: 262144, 6488064, 71368704, 457949184, 1889040384, 5194861056, 9523911936, 11224610496, 7716919716, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+11*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[(4 + 11*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] = (4 + 11*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-a920b56c9b3ef4d1","steps":["basis A013611","reverse"],"edges":["e-551d764375152d3c","e-13b8d53c469bc767"],"text":"A038303 \u2192 basis A013611 \u2192 reverse"},"headline":"A038303 \u2192 basis A013611 \u2192 reverse","chain":null}
