{"id":"n-0e6d847331141ca6","revision":2,"first_index":0,"fingerprint":"f6-f1789e1ed3fa2160","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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"],["2","2"],["3","8","3"],["4","20","20","4"],["5","40","75","40","5"],["6","70","210","210","70","6"],["7","112","490","784","490","112","7"],["8","168","1008","2352","2352","1008","168","8"],["9","240","1890","6048","8820","6048","1890","240","9"],["10","330","3300","13860","27720","27720","13860","3300","330","10"],["11","440","5445","29040","76230","104544","76230","29040","5445","440","11"],["12","572","8580","56628","188760","339768","339768","188760","56628","8580","572","12"],["13","728","13013","104104","429429","981552","1288287","981552","429429","104104","13013","728","13"],["14","910","19110","182182","910910","2576574","4294290","4294290","2576574","910910","182182","19110","910","14"],["15","1120","27300","305760","1821820","6246240","12882870","16359200","12882870","6246240","1821820","305760","27300","1120","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","14","48","165","572","2002","7072","25194","90440","326876","1188640","4345965","15967980","58929450"],"nonn":true,"definitions":[{"position":0,"source":"derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A145596","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","2"],["5","5"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"oeis_catalog:A145596","source_kind":"oeis_catalog","equality_edge_id":"e-34496f1b474c1a7e","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["3","8","3"]],"recurrence":{"denominator":{"coeffs":[["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["7","7"],["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","0","-2"],["-3","-2","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","1","1","-1"],["1","-1","-1","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160365,"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-c483eeee9c2638c9"]},{"id":160366,"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-1d2c7b4a3cb3d72f","e-c483eeee9c2638c9","e-fb63394f9047a486"]},{"id":160367,"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-1d2c7b4a3cb3d72f","e-c483eeee9c2638c9","e-fb63394f9047a486"]},{"id":160368,"position":3,"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-c483eeee9c2638c9"]},{"id":160369,"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-1d2c7b4a3cb3d72f","e-fb63394f9047a486"]},{"id":160370,"position":5,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-fb63394f9047a486"]},{"id":160371,"position":6,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-239cba93e63d8d6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-fb63394f9047a486","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-0e6d847331141ca6","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"42e0016394c21881af94d210212862d15ce1af0a905dd65487a891383b9a265f","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":"unordered"}],"premises":[{"position":0,"node_id":"n-01439b9265fd744f","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}]},{"id":"e-34496f1b474c1a7e","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-0e6d847331141ca6","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":"8b5b921d87c5c66c4a82d19c80443049a3be01f0b288ef23791fdabafd55e633","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-1d2c7b4a3cb3d72f","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-0e6d847331141ca6","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":"a9366cbefdf7b1f4dc43f7386e63c351912980eb22293e5f3f153bd4a65ba19e","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-0e6d847331141ca6","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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-239cba93e63d8d6d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0e6d847331141ca6","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"713432192959747f1bf50dc017352df83bf948d3b71c7769c66b23a553b2e99b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-01439b9265fd744f","role":"sequence","name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]},{"id":"e-c483eeee9c2638c9","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-0e6d847331141ca6","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":"e66fb126b32dd90d30b811c775874c894650d257bd0274a6ce9dfc2da26cd306","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":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-455a924384122a0c","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-449f6ebb0d4b8f65","conclusion_name":"obreschkoff_combination of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-455a924384122a0c","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-449f6ebb0d4b8f65","conclusion_name":"obreschkoff_combination of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-1d2c7b4a3cb3d72f","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-0e6d847331141ca6","conclusion_name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-33d7b30ac5eafba9","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-3c983d83f25bade9","conclusion_name":"shift_x_plus_one of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-3babd9179153850c","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-01439b9265fd744f","conclusion_name":"basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1)."},{"id":"e-915e0a73a4c39c54","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-dd2bce1536f89a0f","conclusion_name":"theta_plus_one_squared of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-92e33d8ed5abb5e6","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-5d9e4549ccbec686","conclusion_name":"one_plus_derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-a1fbec8352aea886","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-d37595541a708e8b","conclusion_name":"mobius_x_over_x_plus_one of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-e77b91d6fbb51ebc","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7bf0ce499426260c","conclusion_name":"derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"},{"id":"e-fabeda6299c4c4d1","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-55791211cdd0b955","conclusion_name":"theta_plus_one of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1"}],"used_by_total":10,"links":[{"id":"l-0e08378d236f8cee","kind":"oeis_match","node_a_id":"n-0e6d847331141ca6","node_b_id":null,"fingerprint":null,"oeis_id":"A002057","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":"A002057","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a3e8918c0ab15999","kind":"oeis_match","node_a_id":"n-0e6d847331141ca6","node_b_id":null,"fingerprint":null,"oeis_id":"A099376","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":"A099376","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-aedf96655ffd0dcb","kind":"oeis_match","node_a_id":"n-0e6d847331141ca6","node_b_id":null,"fingerprint":null,"oeis_id":"A145596","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":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A145596","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fdbe00345c432f34","kind":"oeis_match","node_a_id":"n-0e6d847331141ca6","node_b_id":null,"fingerprint":null,"oeis_id":"A145596","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A145596","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A145596","relation":"identification"},{"oeis_id":"A145596","relation":"match_reversed_triangle"},{"oeis_id":"A002057","relation":"match_row_sums"},{"oeis_id":"A099376","relation":"match_row_sums"},{"oeis_id":"A145596","relation":"match_triangle"}],"bfiles":[{"node_id":"n-0e6d847331141ca6","oeis_id":"A002057","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":12651},{"node_id":"n-0e6d847331141ca6","oeis_id":"A145596","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16256},{"node_id":"n-0e6d847331141ca6","oeis_id":"A145596","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16256}],"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:  2,   2;\n  2:  3,   8,    3;\n  3:  4,  20,   20,     4;\n  4:  5,  40,   75,    40,     5;\n  5:  6,  70,  210,   210,    70,     6;\n  6:  7, 112,  490,   784,   490,   112,     7;\n  7:  8, 168, 1008,  2352,  2352,  1008,   168,    8;\n  8:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;\n  9: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n)*P(n,x) = (n^2+5*n+4+(3*n^2+5*n+2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*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[Together[((n^2 + 5*n + 4 + (3*n^2 + 5*n + 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];\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] = ((n^2 + 5*n + 4 + (3*n^2 + 5*n + 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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":"A145596","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,   2;\n   3:  3,   8,    3;\n   4:  4,  20,   20,     4;\n   5:  5,  40,   75,    40,     5;\n   6:  6,  70,  210,   210,    70,     6;\n   7:  7, 112,  490,   784,   490,   112,     7;\n   8:  8, 168, 1008,  2352,  2352,  1008,   168,    8;\n   9:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;\n  10: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n-3)*P(n,x) = (n^2+3*n+(3*n^2-n)*x)*P(n-1,x) + (2*n*x-2*n*x^2)*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[Together[((n^2 + 3*n + (3*n^2 - n)*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + 3*n + (3*n^2 - n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A145596","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,   2;\n   3:  3,   8,    3;\n   4:  4,  20,   20,     4;\n   5:  5,  40,   75,    40,     5;\n   6:  6,  70,  210,   210,    70,     6;\n   7:  7, 112,  490,   784,   490,   112,     7;\n   8:  8, 168, 1008,  2352,  2352,  1008,   168,    8;\n   9:  9, 240, 1890,  6048,  8820,  6048,  1890,  240,   9;\n  10: 10, 330, 3300, 13860, 27720, 27720, 13860, 3300, 330, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+2*n-3)*P(n,x) = (n^2+3*n+(3*n^2-n)*x)*P(n-1,x) + (2*n*x-2*n*x^2)*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[Together[((n^2 + 3*n + (3*n^2 - n)*x)*P[n - 1, x] + (2*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 + 3*n + (3*n^2 - n)*x)*v[i - 1] + (2*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-0e6d847331141ca6","steps":[],"edges":[],"text":"A145596"},"headline":"A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","chain":null}