{"id":"n-19598058c018a2ec","revision":1,"first_index":0,"fingerprint":"f6-fb13869215ab1dd7","name":"Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].","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"],["1","1"],["3","4","1"],["9","15","7","1"],["27","54","36","10","1"],["81","189","162","66","13","1"],["243","648","675","360","105","16","1"],["729","2187","2673","1755","675","153","19","1"],["2187","7290","10206","7938","3780","1134","210","22","1"],["6561","24057","37908","34020","19278","7182","1764","276","25","1"],["19683","78732","137781","139968","91854","40824","12474","2592","351","28","1"],["59049","255879","492075","557685","415530","214326","78246","20250","3645","435","31","1"],["177147","826686","1732104","2165130","1804275","1058508","449064","138996","31185","4950","528","34","1"],["531441","2657205","6022998","8227494","7577955","4979799","2405700","866052","232551","46035","6534","630","37","1"],["1594323","8503056","20726199","30705480","30961359","22517352","12196899","5003856","1563705","370656","65637","8424","741","40","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","32","128","512","2048","8192","32768","131072","524288","2097152","8388608","33554432","134217728"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A136158","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-1"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169454,"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":false,"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-3af7e4677665f4c0","e-b744eb17b51a23e3"]},{"id":169455,"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-3af7e4677665f4c0","e-aa84cefe7a2893d5","e-b744eb17b51a23e3"]},{"id":169456,"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-3af7e4677665f4c0","e-aa84cefe7a2893d5","e-b744eb17b51a23e3"]},{"id":169457,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-3af7e4677665f4c0"]},{"id":169458,"position":4,"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-aa84cefe7a2893d5"]},{"id":169459,"position":5,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-aa84cefe7a2893d5"]}],"region_exact":{},"incoming_edges":[{"id":"e-3af7e4677665f4c0","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-19598058c018a2ec","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"9349591dc52539444c06ec23daafd9291c9cbc97b2477c4ab9f9d7ad68f9aff6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-aa84cefe7a2893d5","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-19598058c018a2ec","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity","lean_route":null,"lean_declarations":[],"certificate_hash":"f2c435ac37e0492ee1460d57f8fcfded9ab801f58991435639fe2b4fe4989211","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":"previous_interlaces_next"}],"premises":[]},{"id":"e-b744eb17b51a23e3","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-19598058c018a2ec","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"18ed7c27dbee53a9e998294f59e91c5b64b0db193a588bdb4bd6dd61cc2bab37","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-c318ad55d859eb63","kind":"fingerprint_related","node_a_id":"n-685ab5842fbad7f4","node_b_id":"n-19598058c018a2ec","fingerprint":"f6-fb13869215ab1dd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fb13869215ab1dd7"},"other_id":"n-685ab5842fbad7f4","other":{"id":"n-685ab5842fbad7f4","name":"Triangular matrix arising in enumeration of catafusenes, read by rows.","first_index":0,"fingerprint":"f6-fb13869215ab1dd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["1","4","3"]],"card_rows":[["1"],["1","1"],["1","4","3"],["1","7","15","9"]],"row_sums_preview":["1","2","8","32","128","512","2048","8192"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038763"],"derivation":{"root":"n-685ab5842fbad7f4","steps":[],"edges":[],"text":"A038763"},"headline":"A038763: Triangular matrix arising in enumeration of catafusenes, read by rows.","chain":null}},{"id":"l-1d47f19f10a2f501","kind":"oeis_match","node_a_id":"n-19598058c018a2ec","node_b_id":null,"fingerprint":null,"oeis_id":"A136158","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A136158","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2222b5fe4da46a08","kind":"oeis_match","node_a_id":"n-19598058c018a2ec","node_b_id":null,"fingerprint":null,"oeis_id":"A004171","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A004171","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-435cc6fcc990a76a","kind":"oeis_match","node_a_id":"n-19598058c018a2ec","node_b_id":null,"fingerprint":null,"oeis_id":"A038763","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"reversed_triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A038763","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-44a0a03abe79c1c5","kind":"oeis_match","node_a_id":"n-19598058c018a2ec","node_b_id":null,"fingerprint":null,"oeis_id":"A081294","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A081294","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8a8a6115cedca122","kind":"oeis_match","node_a_id":"n-19598058c018a2ec","node_b_id":null,"fingerprint":null,"oeis_id":"A081654","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A081654","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A136158","relation":"identification"},{"oeis_id":"A038763","relation":"match_reversed_triangle"},{"oeis_id":"A004171","relation":"match_row_sums"},{"oeis_id":"A081294","relation":"match_row_sums"},{"oeis_id":"A081654","relation":"match_row_sums"},{"oeis_id":"A136158","relation":"match_triangle"}],"bfiles":[{"node_id":"n-19598058c018a2ec","oeis_id":"A004171","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":13031},{"node_id":"n-19598058c018a2ec","oeis_id":"A038763","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":17215},{"node_id":"n-19598058c018a2ec","oeis_id":"A081294","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":12911},{"node_id":"n-19598058c018a2ec","oeis_id":"A136158","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":17215}],"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:    1,     1;\n  2:    3,     4,     1;\n  3:    9,    15,     7,     1;\n  4:   27,    54,    36,    10,     1;\n  5:   81,   189,   162,    66,    13,    1;\n  6:  243,   648,   675,   360,   105,   16,    1;\n  7:  729,  2187,  2673,  1755,   675,  153,   19,   1;\n  8: 2187,  7290, 10206,  7938,  3780, 1134,  210,  22,  1;\n  9: 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1])/(n - 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}},{"oeis_id":"A038763","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  4,   3;\n  3: 1,  7,  15,    9;\n  4: 1, 10,  36,   54,   27;\n  5: 1, 13,  66,  162,  189,    81;\n  6: 1, 16, 105,  360,  675,   648,   243;\n  7: 1, 19, 153,  675, 1755,  2673,  2187,   729;\n  8: 1, 22, 210, 1134, 3780,  7938, 10206,  7290,  2187;\n  9: 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A136158","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    1,     1;\n  2:    3,     4,     1;\n  3:    9,    15,     7,     1;\n  4:   27,    54,    36,    10,     1;\n  5:   81,   189,   162,    66,    13,    1;\n  6:  243,   648,   675,   360,   105,   16,    1;\n  7:  729,  2187,  2673,  1755,   675,  153,   19,   1;\n  8: 2187,  7290, 10206,  7938,  3780, 1134,  210,  22,  1;\n  9: 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-19598058c018a2ec","steps":[],"edges":[],"text":"A136158"},"headline":"A136158: Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].","chain":null}
