{"id":"n-a13955bc75263637","revision":0,"first_index":1,"fingerprint":"f6-01a7a97b7ee8f4b3","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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"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:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"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":188289,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0004173ef95211f9"]},{"id":188290,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0004173ef95211f9"]},{"id":188291,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0004173ef95211f9"]},{"id":188292,"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-0004173ef95211f9"]}],"region_exact":{},"incoming_edges":[{"id":"e-0004173ef95211f9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-a13955bc75263637","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a7d36b76d4b1f73c02cf44de640024f053899196d4ca9a4cce876120d85e2178","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de5746895014fee3","role":"sequence","name":"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.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4abd4f9fcdbf1c8e","kind":"fingerprint_related","node_a_id":"n-a13955bc75263637","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-8270042dd6030930","other":{"id":"n-8270042dd6030930","name":"basis_map_A008277 of basis_map_A013609 of Triangle of coefficients in expansion of (1+2*x)^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":[["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-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}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        7,          4;\n   3:       49,         72,          16;\n   4:      343,        988,         528,          64;\n   5:     2401,      12240,       11872,        3328,         256;\n   6:    16807,     144244,      227040,      110720,       19200,        1024;\n   7:   117649,    1653912,     3982576,     3011840,      884480,      104448,       4096;\n   8:   823543,   18663628,    66354288,    73155264,    32390400,     6358016,     544768,     16384;\n   9:  5764801,  208594080,  1069968832,  1655367168,  1037600256,   301228032,   42319872,   2752512,    65536;\n  10: 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 >= 2, with P(1,x) = 1.","mathematica":"P[1, 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, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-de5746895014fee3","steps":["D+1"],"edges":["e-0004173ef95211f9"],"text":"A225467 \u2192 D+1"},"headline":"A225467 \u2192 D+1","chain":null}