{"id":"n-f2e89ae5f08e21c7","revision":1,"first_index":0,"fingerprint":"f6-3fe30cac0381d457","name":"(6,1)-Pascal triangle.","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"],["6","1"],["6","7","1"],["6","13","8","1"],["6","19","21","9","1"],["6","25","40","30","10","1"],["6","31","65","70","40","11","1"],["6","37","96","135","110","51","12","1"],["6","43","133","231","245","161","63","13","1"],["6","49","176","364","476","406","224","76","14","1"],["6","55","225","540","840","882","630","300","90","15","1"],["6","61","280","765","1380","1722","1512","930","390","105","16","1"],["6","67","341","1045","2145","3102","3234","2442","1320","495","121","17","1"],["6","73","408","1386","3190","5247","6336","5676","3762","1815","616","138","18","1"],["6","79","481","1794","4576","8437","11583","12012","9438","5577","2431","754","156","19","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","14","28","56","112","224","448","896","1792","3584","7168","14336","28672","57344"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A093563","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169227,"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-3d3bbeacedf897e3","e-4c49a0d263e99a7e"]},{"id":169228,"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-3d3bbeacedf897e3","e-4c49a0d263e99a7e","e-861dfac91b3edbe5"]},{"id":169229,"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-3d3bbeacedf897e3","e-4c49a0d263e99a7e","e-861dfac91b3edbe5"]},{"id":169230,"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-3d3bbeacedf897e3"]},{"id":169231,"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-861dfac91b3edbe5"]},{"id":169232,"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-861dfac91b3edbe5"]}],"region_exact":{},"incoming_edges":[{"id":"e-3d3bbeacedf897e3","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-f2e89ae5f08e21c7","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":"6e79a1d9a308d1e351d6882e1a64d3369c1c3fb14e6e430177e54caa2020c666","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-4c49a0d263e99a7e","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-f2e89ae5f08e21c7","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":"0fa26ae75c568b445a7aebfc3b42fb1ab2fd28ce3545fc35997cda3e04453c77","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]},{"id":"e-861dfac91b3edbe5","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-f2e89ae5f08e21c7","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":"2f08caf5590d53830781edab78867d86d6769a7d007849ecea652e8d15865150","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":[]}],"used_by":[{"id":"e-64c18e8171428bb6","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-e34531ee3663900f","conclusion_name":"basis_map_A013609 of (6,1)-Pascal triangle."},{"id":"e-6cf8ed1a8369900a","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-b707c41a3afefcce","conclusion_name":"basis_map_A013610 of (6,1)-Pascal triangle."},{"id":"e-904624e46717aa64","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-c86f0e2825224734","conclusion_name":"basis_map_A007318 of (6,1)-Pascal triangle."},{"id":"e-a15431d82351f49b","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-f258800410e48031","conclusion_name":"basis_map_A008277 of (6,1)-Pascal triangle."},{"id":"e-ee9fa16dcf9e89d8","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-64a8903e8a3e4c39","conclusion_name":"basis_map_A013611 of (6,1)-Pascal triangle."}],"used_by_total":5,"links":[{"id":"l-cb7a1e92c6ee57ed","kind":"fingerprint_related","node_a_id":"n-f2e89ae5f08e21c7","node_b_id":"n-b58d0e199d27164c","fingerprint":"f6-3fe30cac0381d457","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3fe30cac0381d457"},"other_id":"n-b58d0e199d27164c","other":{"id":"n-b58d0e199d27164c","name":"Pascal (1,6) triangle.","first_index":0,"fingerprint":"f6-3fe30cac0381d457","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["6"],["1","6"],["1","7","6"]],"card_rows":[["6"],["1","6"],["1","7","6"],["1","8","13","6"]],"row_sums_preview":["6","7","14","28","56","112","224","448"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A096956"],"derivation":{"root":"n-b58d0e199d27164c","steps":[],"edges":[],"text":"A096956"},"headline":"A096956: Pascal (1,6) triangle.","chain":null}},{"id":"l-5654bfd645d72f62","kind":"oeis_match","node_a_id":"n-f2e89ae5f08e21c7","node_b_id":null,"fingerprint":null,"oeis_id":"A005009","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":"A005009","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-57a8212405a8bf88","kind":"oeis_match","node_a_id":"n-f2e89ae5f08e21c7","node_b_id":null,"fingerprint":null,"oeis_id":"A135092","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":"A135092","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6e5189538efff8c1","kind":"oeis_match","node_a_id":"n-f2e89ae5f08e21c7","node_b_id":null,"fingerprint":null,"oeis_id":"A093563","in_catalog":1,"target":"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":"triangle","terms_compared":71},"in_catalog":true,"oeis_id":"A093563","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-701c69955d0512c1","kind":"oeis_match","node_a_id":"n-f2e89ae5f08e21c7","node_b_id":null,"fingerprint":null,"oeis_id":"A125176","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":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A125176","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A093563","relation":"identification"},{"oeis_id":"A005009","relation":"match_row_sums"},{"oeis_id":"A125176","relation":"match_row_sums"},{"oeis_id":"A135092","relation":"match_row_sums"},{"oeis_id":"A093563","relation":"match_triangle"}],"bfiles":[{"node_id":"n-f2e89ae5f08e21c7","oeis_id":"A005009","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":7149},{"node_id":"n-f2e89ae5f08e21c7","oeis_id":"A093563","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":71,"oeis_terms_agreeing":71,"oeis_terms_available":71,"term_count":990,"raw_size":11125},{"node_id":"n-f2e89ae5f08e21c7","oeis_id":"A135092","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":33,"oeis_terms_available":33,"term_count":200,"raw_size":7089}],"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: 6,  1;\n  2: 6,  7,   1;\n  3: 6, 13,   8,   1;\n  4: 6, 19,  21,   9,   1;\n  5: 6, 25,  40,  30,  10,   1;\n  6: 6, 31,  65,  70,  40,  11,   1;\n  7: 6, 37,  96, 135, 110,  51,  12,  1;\n  8: 6, 43, 133, 231, 245, 161,  63, 13,  1;\n  9: 6, 49, 176, 364, 476, 406, 224, 76, 14, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (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] = 6 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (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":"A093563","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: 6,  1;\n  2: 6,  7,   1;\n  3: 6, 13,   8,   1;\n  4: 6, 19,  21,   9,   1;\n  5: 6, 25,  40,  30,  10,   1;\n  6: 6, 31,  65,  70,  40,  11,   1;\n  7: 6, 37,  96, 135, 110,  51,  12,  1;\n  8: 6, 43, 133, 231, 245, 161,  63, 13,  1;\n  9: 6, 49, 176, 364, 476, 406, 224, 76, 14, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (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] = 6 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (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-f2e89ae5f08e21c7","steps":[],"edges":[],"text":"A093563"},"headline":"A093563: (6,1)-Pascal triangle.","chain":null}
