{"id":"n-1e2f1b7f566b371c","revision":2,"first_index":0,"fingerprint":"f6-fef35cc74d592534","name":"Pascal-(1,7,1) array.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","9","1"],["1","17","17","1"],["1","25","97","25","1"],["1","33","241","241","33","1"],["1","41","449","1161","449","41","1"],["1","49","721","3297","3297","721","49","1"],["1","57","1057","7161","14721","7161","1057","57","1"],["1","65","1457","13265","44961","44961","13265","1457","65","1"],["1","73","1921","22121","108353","192969","108353","22121","1921","73","1"],["1","81","2449","34241","223329","616049","616049","223329","34241","2449","81","1"],["1","89","3041","50137","412417","1597849","2582881","1597849","412417","50137","3041","89","1"],["1","97","3697","70321","702241","3573569","8493073","8493073","3573569","702241","70321","3697","97","1"],["1","105","4417","95305","1123521","7162729","23251585","35066313","23251585","7162729","1123521","95305","4417","105","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","11","36","149","550","2143","8136","31273","119498","457907","1752300","6709949","25685998","98341639"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A081582","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","7"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:reversal","source_kind":"derived","equality_edge_id":"e-8666650c0d0b3b54","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","1"],["1","-7","-1"],["0","8","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","45","3"],["0","-1","-30","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","30","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158916,"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-a31f5c33917b15a7"]},{"id":158917,"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-7d90ede2da8763e3","e-a31f5c33917b15a7"]},{"id":158918,"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-7d90ede2da8763e3","e-a31f5c33917b15a7"]},{"id":158919,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-a31f5c33917b15a7"]},{"id":158920,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has a real-rooted image"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-7 - sqrt(56), -7 + sqrt(56)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has a real-rooted image","region_exact":null,"edges":["e-216cbdf61e3ed56f"]},{"id":158921,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has an image with all roots in I = [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-7 - sqrt(56), -7 + sqrt(56)] to roots in [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]: every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has an image with all roots in I = [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","region_exact":null,"edges":["e-216cbdf61e3ed56f"]},{"id":158922,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), 0]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every PF input with all roots in [-7 - sqrt(56), 0] has a PF image; no larger [-R, 0] works"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-7 - sqrt(56), 0] to roots in [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]: every PF input with all roots in [-7 - sqrt(56), 0] has a PF image; no larger [-R, 0] works","region_exact":null,"edges":["e-216cbdf61e3ed56f"]},{"id":158923,"position":7,"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-7d90ede2da8763e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-216cbdf61e3ed56f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-1e2f1b7f566b371c","premise_count":0,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"0a6de4c4cd055cfc120b210d9cec81478518f044990b50b6d719f4e0ea460845","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), -7 + sqrt(56)]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every real polynomial with all roots in E = [-7 - sqrt(56), -7 + sqrt(56)] has an image with all roots in I = [-15 - 2 sqrt(56), -15 + 2 sqrt(56)]"},{"claim":"basis_map_preserver","input_interval":"[-7 - sqrt(56), 0]","output_interval":"[-15 - 2 sqrt(56), -15 + 2 sqrt(56)]","statement":"every PF input with all roots in [-7 - sqrt(56), 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},{"id":"e-8666650c0d0b3b54","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-1e2f1b7f566b371c","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"61f0ae813a7583c66165837c9d70f0a810775110ded84258b55db32bca238759","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-7d90ede2da8763e3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-1e2f1b7f566b371c","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"1090e974b89a6db6a15dbbd3229b7c3e410a4defc4e412f257cb278e604d17a1","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-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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]"}]},{"id":"e-a31f5c33917b15a7","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-1e2f1b7f566b371c","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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"17f95e414e52bcda5a737f8000a945a5d6adfca839e8d8425d9e57e1b69df441","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-62ffc182dddb0e7d","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-f7ecea7e153440f6","conclusion_name":"basis_map_A008277 of Pascal-(1,7,1) array."},{"id":"e-85efc751f4220bb8","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-920469c0112e70be","conclusion_name":"basis_map_A013611 of Pascal-(1,7,1) array."},{"id":"e-be875bddda915b22","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-ec7f55e494bf2bbd","conclusion_name":"basis_map_A013610 of Pascal-(1,7,1) array."},{"id":"e-c5fd3da4c5a13f73","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-9f4396728047edfb","conclusion_name":"basis_map_A007318 of Pascal-(1,7,1) array."},{"id":"e-fa71667350448e54","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-351eeaac5d28f739","conclusion_name":"basis_map_A013609 of Pascal-(1,7,1) array."},{"id":"e-c807fb938e5557d1","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-695d6cb541611b1e","conclusion_name":"gamma_polynomial of Pascal-(1,7,1) array."},{"id":"e-03aefd7a03322503","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-9f4396728047edfb","conclusion_name":"basis_map_A007318 of Pascal-(1,7,1) array."},{"id":"e-73cab1ceabab0270","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-ceef1d54dff1e4b1","conclusion_name":"mobius_x_over_x_plus_one of Pascal-(1,7,1) array."},{"id":"e-7d90ede2da8763e3","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-1e2f1b7f566b371c","conclusion_name":"Pascal-(1,7,1) array."},{"id":"e-cc281e0cede67e51","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-fa6c4753786a5757","conclusion_name":"derivative of Pascal-(1,7,1) array."},{"id":"e-e37cbf39b02138a5","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-edb14ad9979fb8e5","conclusion_name":"one_plus_derivative of Pascal-(1,7,1) array."},{"id":"e-ee1a409b71674b97","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-76d229ba9eaa4d3d","conclusion_name":"theta_plus_one of Pascal-(1,7,1) array."},{"id":"e-1c0801c63e403da5","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-691e332e74d300f5","conclusion_name":"veronese_section_2_0 of Pascal-(1,7,1) array."},{"id":"e-bace45a12b7c6e7a","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-632d96a251a43047","conclusion_name":"veronese_section_2_1 of Pascal-(1,7,1) array."}],"used_by_total":14,"links":[{"id":"l-7e2baa64e8d8d290","kind":"oeis_match","node_a_id":"n-1e2f1b7f566b371c","node_b_id":null,"fingerprint":null,"oeis_id":"A081582","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":"A081582","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9d8851faaa4a3ffa","kind":"oeis_match","node_a_id":"n-1e2f1b7f566b371c","node_b_id":null,"fingerprint":null,"oeis_id":"A015519","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A015519","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-da7bffbf18b33093","kind":"oeis_match","node_a_id":"n-1e2f1b7f566b371c","node_b_id":null,"fingerprint":null,"oeis_id":"A081582","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":66},"in_catalog":true,"oeis_id":"A081582","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A081582","relation":"identification"},{"oeis_id":"A081582","relation":"match_reversed_triangle"},{"oeis_id":"A015519","relation":"match_row_sums"},{"oeis_id":"A081582","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1e2f1b7f566b371c","oeis_id":"A081582","target":"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":15567},{"node_id":"n-1e2f1b7f566b371c","oeis_id":"A081582","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":15567}],"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: 1,  9,    1;\n  3: 1, 17,   17,     1;\n  4: 1, 25,   97,    25,     1;\n  5: 1, 33,  241,   241,    33,     1;\n  6: 1, 41,  449,  1161,   449,    41,     1;\n  7: 1, 49,  721,  3297,  3297,   721,    49,    1;\n  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;\n  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,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[(1 + x)*P[n - 1, x] + 7*x*P[n - 2, x]];\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] = (1 + x)*v[i - 1] + 7*x*v[i - 2]); 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":"A081582","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: 1,  9,    1;\n  3: 1, 17,   17,     1;\n  4: 1, 25,   97,    25,     1;\n  5: 1, 33,  241,   241,    33,     1;\n  6: 1, 41,  449,  1161,   449,    41,     1;\n  7: 1, 49,  721,  3297,  3297,   721,    49,    1;\n  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;\n  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,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[(1 + x)*P[n - 1, x] + 7*x*P[n - 2, x]];\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] = (1 + x)*v[i - 1] + 7*x*v[i - 2]); 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}},{"oeis_id":"A081582","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,  9,    1;\n  3: 1, 17,   17,     1;\n  4: 1, 25,   97,    25,     1;\n  5: 1, 33,  241,   241,    33,     1;\n  6: 1, 41,  449,  1161,   449,    41,     1;\n  7: 1, 49,  721,  3297,  3297,   721,    49,    1;\n  8: 1, 57, 1057,  7161, 14721,  7161,  1057,   57,  1;\n  9: 1, 65, 1457, 13265, 44961, 44961, 13265, 1457, 65, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + 7*x*P(n-2,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[(1 + x)*P[n - 1, x] + 7*x*P[n - 2, x]];\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] = (1 + x)*v[i - 1] + 7*x*v[i - 2]); 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}}],"derivation":{"root":"n-1e2f1b7f566b371c","steps":[],"edges":[],"text":"A081582"},"headline":"A081582: Pascal-(1,7,1) array.","chain":null}