{"id":"n-301221ae3d9e17f8","revision":0,"first_index":0,"fingerprint":"f6-322b0642acfc2857","name":"basis_map_A008277 of basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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":false,"rows":[["1"],["3","1"],["18","14","2"],["162","222","72","6"],["1944","4200","2328","432","24"],["29160","93720","79200","24600","3000","120"],["524880","2424240","2938320","1360800","273600","23760","720"],["11022480","71552880","119810880","77721840","22932000","3245760","211680","5040"],["264539520","2377872000","5364858240","4689135360","1905966720","391184640","41207040","2096640","40320"],["7142567040","87984247680","262819468800","301497033600","162278121600","45318994560","6858432000","559560960","22861440","362880"],["214277011200","3590795577600","14020815916800","20718016704000","14374438848000","5248300780800","1070448825600","124540416000","8110368000","272160000","3628800"],["7071141369600","160352052537600","810643626777600","1521618077548800","1334729975040000","619969296038400","163705742323200","25474382841600","2351299104000","125139168000","3512678400","39916800"],["254561089305600","7781752218240000","50562842237107200","119284225104844800","130376734833945600","75533812120166400","25119851723366400","5021394848870400","616064075827200","46235629440000","2049647846400","48858163200","479001600"],["9927882482918400","407960409509452800","3387747524249088000","9961486507260288000","13414977796252032000","9550414013338598400","3920962209196032000","979339402756915200","153375295878604800","15221591178393600","947453668761600","35537607705600","728561433600","6227020800"],["416971064282572800","22984773287290214400","242853772430568499200","884193331949351424000","1454128635134343168000","1257456058641451468800","627747034545440409600","192000987314712576000","37330547203901952000","4704481460270592000","385538844212121600","20229722472960000","650524408934400","11594712729600","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","34","462","8928","229800","7546320","306502560","15036900480","874481650560","59370020236800","4646044285344000","414599572866124800","41802396358167244800","4724223023804597913600"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197521,"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-bb568b842b6494ed"]},{"id":197522,"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-bb568b842b6494ed"]},{"id":197523,"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-bb568b842b6494ed"]}],"region_exact":{},"incoming_edges":[{"id":"e-bb568b842b6494ed","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-301221ae3d9e17f8","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"6743c27994cabfca2120aab2004d64cf8f67e32507c691e038e7a2b1d7f40045","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-17166a4ad91393c8","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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-1248539bc6407997","kind":"fingerprint_related","node_a_id":"n-a2b8ff03da0c14a3","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-a2b8ff03da0c14a3","other":{"id":"n-a2b8ff03da0c14a3","name":"Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["9","7","1"]],"card_rows":[["1"],["3","1"],["9","7","1"],["27","37","12","1"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A143495"],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":[],"edges":[],"text":"A143495"},"headline":"A143495: Triangle read by rows: 3-Stirling numbers of the second kind.","chain":null}},{"id":"l-2adfec560157ece3","kind":"fingerprint_related","node_a_id":"n-17e55d1ba7fd6f6e","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-17e55d1ba7fd6f6e","other":{"id":"n-17e55d1ba7fd6f6e","name":"basis_map_A008277 of basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["15","5"],["135","105","15"]],"card_rows":[["1"],["15","5"],["135","105","15"],["945","1295","420","35"]],"row_sums_preview":["1","20","255","2695","26040","241290","2201010","20068620"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318","basis A008277"],"edges":["e-484d449f4d0f99c5","e-c70ca5801cc82715"],"text":"A178821 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A178821 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-2d14a396b84846d7","kind":"fingerprint_related","node_a_id":"n-ad42851570f8c7d6","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-ad42851570f8c7d6","other":{"id":"n-ad42851570f8c7d6","name":"basis_map_A008277 of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["18","14","2"]],"card_rows":[["1"],["3","1"],["18","14","2"],["135","185","60","5"]],"row_sums_preview":["1","4","34","385","5208","80430","1383492","26089206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-45645d17dad7cf5c","e-7283a15d3aaeffa5"],"text":"A085880 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A085880 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-33fabd885b7ed33c","kind":"fingerprint_related","node_a_id":"n-301221ae3d9e17f8","node_b_id":"n-34825491574a3572","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-34825491574a3572","other":{"id":"n-34825491574a3572","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","4"],["144","112","16"]],"card_rows":[["1"],["12","4"],["144","112","16"],["1728","2368","768","64"]],"row_sums_preview":["1","16","272","4928","95232","1960960","42930176","996376576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611","basis A008277"],"edges":["e-267942f18014ef6e","e-bfbf50a574fa76ff"],"text":"A027466 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A027466 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-34a0a07bfe2a2686","kind":"fingerprint_related","node_a_id":"n-69f36ca70371cbe2","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-69f36ca70371cbe2","other":{"id":"n-69f36ca70371cbe2","name":"basis_map_A008277 of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","4"],["90","70","10"]],"card_rows":[["1"],["12","4"],["90","70","10"],["540","740","240","20"]],"row_sums_preview":["1","16","170","1540","13020","107240","880404","7297680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318","basis A008277"],"edges":["e-2fa6de2302a1c958","e-e38049b16ce90a7b"],"text":"A178820 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A178820 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-36df8bd3a88109b4","kind":"fingerprint_related","node_a_id":"n-19894faf8daea03d","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-19894faf8daea03d","other":{"id":"n-19894faf8daea03d","name":"basis_map_A008277 of shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["27","21","3"],["108","148","48","4"]],"row_sums_preview":["0","1","8","51","308","1860","11490","73367"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-1a916fd0c60fb95f","e-ca25fa36bbb89e58"],"text":"A003506 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A003506 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-44b2416dbd53c7c2","kind":"fingerprint_related","node_a_id":"n-301221ae3d9e17f8","node_b_id":"n-5409544d947a9f9b","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-5409544d947a9f9b","other":{"id":"n-5409544d947a9f9b","name":"basis_map_A008277 of basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["33","11"],["1089","847","121"]],"card_rows":[["1"],["33","11"],["1089","847","121"],["35937","49247","15972","1331"]],"row_sums_preview":["1","44","2057","102487","5446452","308412665","18567730841","1185092817194"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318","basis A008277"],"edges":["e-a5d0659f5315d3b7","e-0580db642f73c65b"],"text":"A038325 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A038325 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-44d2c0937d74b765","kind":"fingerprint_related","node_a_id":"n-40f55dba6bf772b4","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-40f55dba6bf772b4","other":{"id":"n-40f55dba6bf772b4","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["9","7","1"]],"card_rows":[["1"],["3","1"],["9","7","1"],["27","37","12","1"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["basis A008277"],"edges":["e-74a6e78fdd5274b8"],"text":"A038207 \u2192 basis A008277"},"headline":"A038207 \u2192 basis A008277","chain":null}},{"id":"l-58d4e10b84ff142f","kind":"fingerprint_related","node_a_id":"n-06404fc73ac5a6a1","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-06404fc73ac5a6a1","other":{"id":"n-06404fc73ac5a6a1","name":"reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","9"]],"card_rows":[["1"],["1","3"],["1","7","9"],["1","12","37","27"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["reverse"],"edges":["e-0c475173369de966"],"text":"A143495 \u2192 reverse"},"headline":"A143495 \u2192 reverse","chain":null}},{"id":"l-6ed446b7a1706c05","kind":"fingerprint_related","node_a_id":"n-301221ae3d9e17f8","node_b_id":"n-f67f18b926dba7e3","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-f67f18b926dba7e3","other":{"id":"n-f67f18b926dba7e3","name":"basis_map_A008277 of basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["27","21","3"]],"card_rows":[["1"],["3","1"],["27","21","3"],["405","555","180","15"]],"row_sums_preview":["1","4","51","1155","39060","1809675","108949995","8218099890"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318","basis A008277"],"edges":["e-5048ce3ebd740a6f","e-ce1a4eb45988cf81"],"text":"A085881 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A085881 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-73c8e27fd3f8ed82","kind":"fingerprint_related","node_a_id":"n-7f135bf281132ad6","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-7f135bf281132ad6","other":{"id":"n-7f135bf281132ad6","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","3"],["81","63","9"]],"card_rows":[["1"],["9","3"],["81","63","9"],["729","999","324","27"]],"row_sums_preview":["1","12","153","2079","30132","465345","7640649","133000218"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-02e8db4659f506e0","e-aee5977c1fa2e0a2"],"text":"A038221 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A038221 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-7c66fbfeaeeb2fbe","kind":"fingerprint_related","node_a_id":"n-301221ae3d9e17f8","node_b_id":"n-38c44069a108b9ce","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-38c44069a108b9ce","other":{"id":"n-38c44069a108b9ce","name":"basis_map_A008277 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["576","448","64"],["13824","18944","6144","512"],["331776","716800","397312","73728","4096"]],"card_rows":[["576","448","64"],["13824","18944","6144","512"],["331776","716800","397312","73728","4096"],["7962624","25591808","21626880","6717440","819200","32768"]],"row_sums_preview":["1088","39424","1523712","62750720","2747531264","127536201728","6256878157824","323442846990336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609","basis A008277"],"edges":["e-6efc8fa5932928ce","e-4b8bf108795c7bf5"],"text":"A038330 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A038330 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-84502eb9674e81a7","kind":"fingerprint_related","node_a_id":"n-42ff5484acef49d8","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-42ff5484acef49d8","other":{"id":"n-42ff5484acef49d8","name":"basis_map_A008277 of basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["54","42","6"]],"card_rows":[["1"],["6","2"],["54","42","6"],["540","740","240","20"]],"row_sums_preview":["1","8","102","1540","26040","482580","9684444","208713648"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-0f8327ed3206c9f7","e-a0feed404d681466"],"text":"A356546 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A356546 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-8c2dcfec748a7acb","kind":"fingerprint_related","node_a_id":"n-0446d4cb1fe0a9ca","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-0446d4cb1fe0a9ca","other":{"id":"n-0446d4cb1fe0a9ca","name":"basis_map_A008277 of shift_x_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","3"],["54","42","6"]],"card_rows":[["1"],["9","3"],["54","42","6"],["270","370","120","10"]],"row_sums_preview":["1","12","102","770","5580","40215","293468","2189304"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-34e3d9a08d46fb54","e-abf44acaba5824db"],"text":"A094305 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A094305 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-8fd19757ab21062e","kind":"fingerprint_related","node_a_id":"n-3f70e5fe46f2ce5d","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-3f70e5fe46f2ce5d","other":{"id":"n-3f70e5fe46f2ce5d","name":"basis_map_A008277 of shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-3","-1"],["9","7","1"]],"card_rows":[["1"],["-3","-1"],["9","7","1"],["-27","-37","-12","-1"]],"row_sums_preview":["1","-4","17","-77","372","-1915","10481","-60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-51a320c1f4e00092","e-d060ffbe5cea159c"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-cb6bf9a5e5fb70b8","kind":"fingerprint_related","node_a_id":"n-ffc3f9478b0d8c7f","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-ffc3f9478b0d8c7f","other":{"id":"n-ffc3f9478b0d8c7f","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["36","28","4"]],"card_rows":[["1"],["6","2"],["36","28","4"],["216","296","96","8"]],"row_sums_preview":["1","8","68","616","5952","61280","670784","7784192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318","basis A008277"],"edges":["e-1756f066a00732e3","e-3f600b740c1073d4"],"text":"A038208 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A038208 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-cbfaeeb7662111fc","kind":"fingerprint_related","node_a_id":"n-c823eaf24c675454","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-c823eaf24c675454","other":{"id":"n-c823eaf24c675454","name":"basis_map_A008277 of shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","4"],["81","63","9"]],"card_rows":[["1"],["12","4"],["81","63","9"],["432","592","192","16"]],"row_sums_preview":["1","16","153","1232","9300","68940","513569","3892096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-18ace392e620d7bd","e-1a326128b4362dc4"],"text":"A135065 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A135065 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-d0f643d43c5765cc","kind":"fingerprint_related","node_a_id":"n-6b58df8d9b432203","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-6b58df8d9b432203","other":{"id":"n-6b58df8d9b432203","name":"basis_map_A008277 of derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["18","6"]],"card_rows":[["3"],["18","6"],["81","63","9"],["324","444","144","12"]],"row_sums_preview":["0","3","24","153","924","5580","34470","220101"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D","basis A008277"],"edges":["e-d19cf3982d12ec2e","e-6b011e329e1f2f47"],"text":"A120910 \u2192 D \u2192 basis A008277"},"headline":"A120910 \u2192 D \u2192 basis A008277","chain":null}},{"id":"l-d1230f4718ce14ca","kind":"fingerprint_related","node_a_id":"n-0c5bef71f8e4beb5","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-0c5bef71f8e4beb5","other":{"id":"n-0c5bef71f8e4beb5","name":"basis_map_A008277 of basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["18","6"],["189","147","21"]],"card_rows":[["1"],["18","6"],["189","147","21"],["1512","2072","672","56"]],"row_sums_preview":["1","24","357","4312","46872","482580","4842222","48164688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-a52727d7b51dd683","e-be86b9a472a5134b"],"text":"A178822 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A178822 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-d3b1e42a426d01c5","kind":"fingerprint_related","node_a_id":"n-9c4a5ff282fc23ef","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-9c4a5ff282fc23ef","other":{"id":"n-9c4a5ff282fc23ef","name":"reversal of basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","9"]],"card_rows":[["1"],["1","3"],["1","7","9"],["1","12","37","27"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["basis A008277","reverse"],"edges":["e-74a6e78fdd5274b8","e-48d2e21c9668fe16"],"text":"A038207 \u2192 basis A008277 \u2192 reverse"},"headline":"A038207 \u2192 basis A008277 \u2192 reverse","chain":null}},{"id":"l-dbcc9f9b9b78d616","kind":"fingerprint_related","node_a_id":"n-301221ae3d9e17f8","node_b_id":"n-c0f3c67429e9bec7","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-c0f3c67429e9bec7","other":{"id":"n-c0f3c67429e9bec7","name":"basis_map_A008277 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1296","1008","144"],["46656","63936","20736","1728"],["1679616","3628800","2011392","373248","20736"]],"card_rows":[["1296","1008","144"],["46656","63936","20736","1728"],["1679616","3628800","2011392","373248","20736"],["60466176","194337792","164229120","51010560","6220800","248832"]],"row_sums_preview":["2448","133056","7713792","476513280","31296098304","2179075571712","160356943724544","12434229604122624"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-240272ddc3b4d83d","e-60262c556dac21ec"],"text":"A038338 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A038338 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}},{"id":"l-eddd2e65c5a4803c","kind":"fingerprint_related","node_a_id":"n-4755d98ca7204416","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-4755d98ca7204416","other":{"id":"n-4755d98ca7204416","name":"basis_map_A008277 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["9","3"],["27","21","3"]],"card_rows":[["3"],["9","3"],["27","21","3"],["81","111","36","3"]],"row_sums_preview":["3","12","51","231","1116","5745","31443","182442"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A008277"],"edges":["e-056f2aef56ba708a"],"text":"A120910 \u2192 basis A008277"},"headline":"A120910 \u2192 basis A008277","chain":null}},{"id":"l-ee1671a20fba3944","kind":"fingerprint_related","node_a_id":"n-5bd42e161dd707c1","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-5bd42e161dd707c1","other":{"id":"n-5bd42e161dd707c1","name":"one_plus_derivative of basis_map_A008277 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["54","42","6"]],"card_rows":[["1"],["6","2"],["54","42","6"],["648","888","288","24"]],"row_sums_preview":["1","8","102","1848","44640","1378800","52824240","2452020480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318","basis A008277"],"edges":["e-bd56c468ffc76132","e-d0627c8203c99c93"],"text":"A156992 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A156992 \u2192 basis A007318 \u2192 basis A008277","chain":null}},{"id":"l-fd3dd34923e341d5","kind":"fingerprint_related","node_a_id":"n-862a8fa15cf2d66b","node_b_id":"n-301221ae3d9e17f8","fingerprint":"f6-322b0642acfc2857","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-322b0642acfc2857"},"other_id":"n-862a8fa15cf2d66b","other":{"id":"n-862a8fa15cf2d66b","name":"basis_map_A008277 of shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-322b0642acfc2857","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["6","2"],["18","14","2"]],"card_rows":[["2"],["6","2"],["18","14","2"],["54","74","24","2"]],"row_sums_preview":["2","8","34","154","744","3830","20962","121628"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318","basis A008277"],"edges":["e-014dd8a0f9594ad5","e-a0e1d232018924e7"],"text":"A028326 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A028326 \u2192 basis A007318 \u2192 basis A008277","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,           1;\n  2:         18,          14,            2;\n  3:        162,         222,           72,            6;\n  4:       1944,        4200,         2328,          432,           24;\n  5:      29160,       93720,        79200,        24600,         3000,         120;\n  6:     524880,     2424240,      2938320,      1360800,       273600,       23760,        720;\n  7:   11022480,    71552880,    119810880,     77721840,     22932000,     3245760,     211680,      5040;\n  8:  264539520,  2377872000,   5364858240,   4689135360,   1905966720,   391184640,   41207040,   2096640,    40320;\n  9: 7142567040, 87984247680, 262819468800, 301497033600, 162278121600, 45318994560, 6858432000, 559560960, 22861440, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], 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] = (3*n + n*x)*v[i - 1] + n*x*deriv(v[i - 1], x)); 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-cb5d49e71e46f2ed","steps":["x \u21a6 x+1","basis A008277"],"edges":["e-0b89a3d3e0cfd72c","e-bb568b842b6494ed"],"text":"A196347 \u2192 x \u21a6 x+1 \u2192 basis A008277"},"headline":"A196347 \u2192 x \u21a6 x+1 \u2192 basis A008277","chain":null}