{"id":"n-8270042dd6030930","revision":0,"first_index":0,"fingerprint":"f6-01a7a97b7ee8f4b3","name":"basis_map_A008277 of basis_map_A013609 of Triangle of coefficients in expansion of (1+2*x)^n.","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"],["7","4"],["49","72","16"],["343","988","528","64"],["2401","12240","11872","3328","256"],["16807","144244","227040","110720","19200","1024"],["117649","1653912","3982576","3011840","884480","104448","4096"],["823543","18663628","66354288","73155264","32390400","6358016","544768","16384"],["5764801","208594080","1069968832","1655367168","1037600256","301228032","42319872","2752512","65536"],["40353607","2317594084","16883908800","35731851520","30486274560","12283557888","2516828160","265617408","13565952","262144"],["282475249","25654949352","262529008336","746440814080","844111720960","453601161216","127155904512","19363921920","1591541760","65536000","1048576"],["1977326743","283334343868","4040554922448","15232491500864","22400332838400","15623678236672","5756237684736","1186360885248","139525816320","9184215040","311427072","4194304"],["13841287201","3124587089520","61741661212192","305579558206208","576137621286656","511440643743744","240938081173504","64547581722624","10186950377472","953024512000","51373932544","1459617792","16777216"],["96889010407","34425823133524","938623266540960","6052978250766720","14473483522417920","16113447866227712","9514843091353600","3222917684985856","655481391611904","81727855525888","6226672877568","279936237568","6761218048","67108864"],["678223072849","379071610510392","14217052290648496","118761079830731520","357102034018679040","492957026477819904","359413927296872448","150861491339919360","38455445012807680","6136223354060800","619565047349248","39183439626240","1491611942912","31004295168","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","137","1923","30097","519035","9759001","198306291","4323661089","100539814123","2480798081961","64673993391715","1774716400938673","51094539079016539","1538944300826771513"],"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":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197101,"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-e4aa5d8115f76f10"]},{"id":197102,"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-e4aa5d8115f76f10"]},{"id":197103,"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-e4aa5d8115f76f10"]}],"region_exact":{},"incoming_edges":[{"id":"e-e4aa5d8115f76f10","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8270042dd6030930","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":"6c8caac8c33379004e1406317e4c0284ef55dad186cfaba89e5f43f3a9884205","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-026074a09c74735a","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+2*x)^n.","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-8204d3b6e000456a","kind":"fingerprint_related","node_a_id":"n-8270042dd6030930","node_b_id":"n-6c7b3ea8bb66c042","fingerprint":"f6-01a7a97b7ee8f4b3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01a7a97b7ee8f4b3"},"other_id":"n-6c7b3ea8bb66c042","other":{"id":"n-6c7b3ea8bb66c042","name":"basis_map_A008277 of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-01a7a97b7ee8f4b3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["21","12"],["147","216","48"]],"card_rows":[["3"],["21","12"],["147","216","48"],["1029","2964","1584","192"]],"row_sums_preview":["3","33","411","5769","90291","1557105","29277003","594918873"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013611","basis A008277"],"edges":["e-1a05d30902084644","e-7a9632536d668834"],"text":"A120910 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A120910 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-9a23ec3bd2c6e590","kind":"fingerprint_related","node_a_id":"n-a13955bc75263637","node_b_id":"n-8270042dd6030930","fingerprint":"f6-01a7a97b7ee8f4b3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01a7a97b7ee8f4b3"},"other_id":"n-a13955bc75263637","other":{"id":"n-a13955bc75263637","name":"one_plus_derivative of Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","first_index":1,"fingerprint":"f6-01a7a97b7ee8f4b3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","4"],["49","72","16"]],"card_rows":[["1"],["7","4"],["49","72","16"],["343","988","528","64"]],"row_sums_preview":["1","11","137","1923","30097","519035","9759001","198306291"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-de5746895014fee3","steps":["D+1"],"edges":["e-0004173ef95211f9"],"text":"A225467 \u2192 D+1"},"headline":"A225467 \u2192 D+1","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:        7,          4;\n  2:       49,         72,          16;\n  3:      343,        988,         528,          64;\n  4:     2401,      12240,       11872,        3328,         256;\n  5:    16807,     144244,      227040,      110720,       19200,        1024;\n  6:   117649,    1653912,     3982576,     3011840,      884480,      104448,       4096;\n  7:   823543,   18663628,    66354288,    73155264,    32390400,     6358016,     544768,     16384;\n  8:  5764801,  208594080,  1069968832,  1655367168,  1037600256,   301228032,   42319872,   2752512,    65536;\n  9: 40353607, 2317594084, 16883908800, 35731851520, 30486274560, 12283557888, 2516828160, 265617408, 13565952, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) + 4*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[(7 + 4*x)*P[n - 1, x] + 4*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] = (7 + 4*x)*v[i - 1] + 4*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-71965e79e47a4526","steps":["basis A013609","basis A008277"],"edges":["e-95924f9cf72f1e84","e-e4aa5d8115f76f10"],"text":"A013609 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A013609 \u2192 basis A013609 \u2192 basis A008277","chain":null}