{"id":"n-d8cfb1e39c5d56b0","revision":1,"first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","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"],["1","1"],["1","3","1"],["1","5","6","1"],["1","7","15","10","1"],["1","9","28","35","15","1"],["1","11","45","84","70","21","1"],["1","13","66","165","210","126","28","1"],["1","15","91","286","495","462","210","36","1"],["1","17","120","455","1001","1287","924","330","45","1"],["1","19","153","680","1820","3003","3003","1716","495","55","1"],["1","21","190","969","3060","6188","8008","6435","3003","715","66","1"],["1","23","231","1330","4845","11628","18564","19448","12870","5005","1001","78","1"],["1","25","276","1771","7315","20349","38760","50388","43758","24310","8008","1365","91","1"],["1","27","325","2300","10626","33649","74613","116280","125970","92378","43758","12376","1820","105","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","5","13","34","89","233","610","1597","4181","10946","28657","75025","196418","514229"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-3","-11"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161350,"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-6116c33129f68e6c"]},{"id":161351,"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-6116c33129f68e6c"]},{"id":161352,"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-6116c33129f68e6c"]}],"region_exact":{},"incoming_edges":[{"id":"e-6116c33129f68e6c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-d8cfb1e39c5d56b0","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":"6f94f6a9c02e9cfee781dab762f683dc963c1f5e366c82eb2cbb33f5c93876c8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0eba401b117f16a9","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","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-1f6c5defda3a35ef","kind":"fingerprint_related","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":"n-4c232e962100ae75","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-4c232e962100ae75","other":{"id":"n-4c232e962100ae75","name":"reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6758fe7729845be8","steps":["reverse"],"edges":["e-99c1ee54068faff3"],"text":"A054142 \u2192 reverse"},"headline":"A054142 \u2192 reverse","chain":null}},{"id":"l-243352c63191b0ef","kind":"fingerprint_related","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":"n-ff36538314ab71f5","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-ff36538314ab71f5","other":{"id":"n-ff36538314ab71f5","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","48","16"]],"card_rows":[["1"],["4","4"],["16","48","16"],["64","320","384","64"]],"row_sums_preview":["1","8","80","832","8704","91136","954368","9994240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611"],"edges":["e-36be6b9e5c42de00"],"text":"A103327 \u2192 basis A013611"},"headline":"A103327 \u2192 basis A013611","chain":null}},{"id":"l-6edf36d7ad5e7467","kind":"fingerprint_related","node_a_id":"n-01fe968f3e05a927","node_b_id":"n-d8cfb1e39c5d56b0","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-01fe968f3e05a927","other":{"id":"n-01fe968f3e05a927","name":"Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","1"]],"row_sums_preview":["1","1","2","5","13","34","89","233"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)"],"region_exact":{},"oeis_ids":["A165253"],"derivation":{"root":"n-01fe968f3e05a927","steps":[],"edges":[],"text":"A165253"},"headline":"A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","chain":null}},{"id":"l-87bc6e849531e4cd","kind":"fingerprint_related","node_a_id":"n-0eba401b117f16a9","node_b_id":"n-d8cfb1e39c5d56b0","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-0eba401b117f16a9","other":{"id":"n-0eba401b117f16a9","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)"],"region_exact":{},"oeis_ids":["A085478"],"derivation":{"root":"n-0eba401b117f16a9","steps":[],"edges":[],"text":"A085478"},"headline":"A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","chain":null}},{"id":"l-a44e611149cd9c68","kind":"fingerprint_related","node_a_id":"n-6758fe7729845be8","node_b_id":"n-d8cfb1e39c5d56b0","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-6758fe7729845be8","other":{"id":"n-6758fe7729845be8","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","5","6","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A054142"],"derivation":{"root":"n-6758fe7729845be8","steps":[],"edges":[],"text":"A054142"},"headline":"A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","chain":null}},{"id":"l-61ef5c222e537a17","kind":"oeis_match","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":null,"fingerprint":null,"oeis_id":"A048575","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":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A048575","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6a6f322c5ccaf0dd","kind":"oeis_match","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":null,"fingerprint":null,"oeis_id":"A001519","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001519","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a313eacac1a59df7","kind":"oeis_match","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":null,"fingerprint":null,"oeis_id":"A122367","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":15},"in_catalog":false,"oeis_id":"A122367","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d2116934bdce431c","kind":"oeis_match","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":null,"fingerprint":null,"oeis_id":"A085478","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"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":"A085478","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f970775fdfb56895","kind":"oeis_match","node_a_id":"n-d8cfb1e39c5d56b0","node_b_id":null,"fingerprint":null,"oeis_id":"A054142","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"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":"A054142","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A085478","relation":"match_reversed_triangle"},{"oeis_id":"A001519","relation":"match_row_sums"},{"oeis_id":"A048575","relation":"match_row_sums"},{"oeis_id":"A122367","relation":"match_row_sums"},{"oeis_id":"A054142","relation":"match_triangle"}],"bfiles":[{"node_id":"n-d8cfb1e39c5d56b0","oeis_id":"A048575","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":9364},{"node_id":"n-d8cfb1e39c5d56b0","oeis_id":"A054142","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":12893},{"node_id":"n-d8cfb1e39c5d56b0","oeis_id":"A085478","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":12893},{"node_id":"n-d8cfb1e39c5d56b0","oeis_id":"A122367","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":9281}],"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: 1,  1;\n   3: 1,  3,   1;\n   4: 1,  5,   6,   1;\n   5: 1,  7,  15,  10,    1;\n   6: 1,  9,  28,  35,   15,    1;\n   7: 1, 11,  45,  84,   70,   21,   1;\n   8: 1, 13,  66, 165,  210,  126,  28,   1;\n   9: 1, 15,  91, 286,  495,  462, 210,  36,  1;\n  10: 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = (2*n-3+(6*n-11)*x)*P(n-1,x) - (x+4*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[((2*n - 3 + (6*n - 11)*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\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] = ((2*n - 3 + (6*n - 11)*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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}},{"oeis_id":"A054142","target":"triangle","first_row":1,"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,  3,   1;\n  3: 1,  5,   6,   1;\n  4: 1,  7,  15,  10,    1;\n  5: 1,  9,  28,  35,   15,    1;\n  6: 1, 11,  45,  84,   70,   21,   1;\n  7: 1, 13,  66, 165,  210,  126,  28,   1;\n  8: 1, 15,  91, 286,  495,  462, 210,  36,  1;\n  9: 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n-1+(6*n-5)*x)*P(n-1,x) - (x+4*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[((2*n - 1 + (6*n - 5)*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x])/(2*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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n - 1 + (6*n - 5)*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x))/(2*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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A085478","target":"reversed_triangle","first_row":1,"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,  3,   1;\n  3: 1,  6,   5,   1;\n  4: 1, 10,  15,   7,    1;\n  5: 1, 15,  35,  28,    9,    1;\n  6: 1, 21,  70,  84,   45,   11,   1;\n  7: 1, 28, 126, 210,  165,   66,  13,   1;\n  8: 1, 36, 210, 462,  495,  286,  91,  15,  1;\n  9: 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1)*P(n,x) = (2*n-1+(6*n-5)*x)*P(n-1,x) - (x+4*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[((2*n - 1 + (6*n - 5)*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x])/(2*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; for(n = 1, N, my(i = n + 1); v[i] = ((2*n - 1 + (6*n - 5)*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x))/(2*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":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-0eba401b117f16a9","steps":["reverse"],"edges":["e-6116c33129f68e6c"],"text":"A085478 \u2192 reverse"},"headline":"A085478 \u2192 reverse","chain":null}
