{"id":"n-6fa6977267f2e73d","revision":0,"first_index":0,"fingerprint":"f6-d272e4b2958d4a94","name":"Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).","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":false,"rows":[["1"],["0","1"],["0","2","1"],["0","0","6","1"],["0","0","6","12","1"],["0","0","0","30","20","1"],["0","0","0","20","90","30","1"],["0","0","0","0","140","210","42","1"],["0","0","0","0","70","560","420","56","1"],["0","0","0","0","0","630","1680","756","72","1"],["0","0","0","0","0","252","3150","4200","1260","90","1"],["0","0","0","0","0","0","2772","11550","9240","1980","110","1"],["0","0","0","0","0","0","924","16632","34650","18480","2970","132","1"],["0","0","0","0","0","0","0","12012","72072","90090","34320","4290","156","1"],["0","0","0","0","0","0","0","3432","84084","252252","210210","60060","6006","182","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","3","7","19","51","141","393","1107","3139","8953","25653","73789","212941","616227"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A105868","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","1"],["0","4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169312,"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":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-e3024342b1585d9b"]},{"id":169313,"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-e3024342b1585d9b"]},{"id":169314,"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-e3024342b1585d9b"]}],"region_exact":{},"incoming_edges":[{"id":"e-e3024342b1585d9b","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-6fa6977267f2e73d","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"16241fad6da82565f0b788680b7e4125216e9fcb478c6dea88811934733b32c6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-344d646a91640f03","kind":"fingerprint_related","node_a_id":"n-8bfdd58279be42d7","node_b_id":"n-6fa6977267f2e73d","fingerprint":"f6-d272e4b2958d4a94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d272e4b2958d4a94"},"other_id":"n-8bfdd58279be42d7","other":{"id":"n-8bfdd58279be42d7","name":"Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","first_index":0,"fingerprint":"f6-d272e4b2958d4a94","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","6"]],"row_sums_preview":["1","1","3","7","19","51","141","393"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A391403"],"derivation":{"root":"n-8bfdd58279be42d7","steps":[],"edges":[],"text":"A391403"},"headline":"A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","chain":null}},{"id":"l-36715448f106c652","kind":"fingerprint_related","node_a_id":"n-ce7f61ca1eb91390","node_b_id":"n-6fa6977267f2e73d","fingerprint":"f6-d272e4b2958d4a94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d272e4b2958d4a94"},"other_id":"n-ce7f61ca1eb91390","other":{"id":"n-ce7f61ca1eb91390","name":"gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-d272e4b2958d4a94","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["6","12"]],"card_rows":[["1"],["2"],["6","12"],["20","120"]],"row_sums_preview":["1","2","18","140","1330","12852","130284","1348776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["\u03b3"],"edges":["e-510754e51c3ada00"],"text":"A069466 \u2192 \u03b3"},"headline":"A069466 \u2192 \u03b3","chain":null}},{"id":"l-62467b60739604bb","kind":"fingerprint_related","node_a_id":"n-4b168caf0bc63f8b","node_b_id":"n-6fa6977267f2e73d","fingerprint":"f6-d272e4b2958d4a94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d272e4b2958d4a94"},"other_id":"n-4b168caf0bc63f8b","other":{"id":"n-4b168caf0bc63f8b","name":"T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-d272e4b2958d4a94","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","6"]],"row_sums_preview":["1","1","3","7","19","51","141","393"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A089627"],"derivation":{"root":"n-4b168caf0bc63f8b","steps":[],"edges":[],"text":"A089627"},"headline":"A089627: T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","chain":null}},{"id":"l-2f40432cec31a0c1","kind":"oeis_match","node_a_id":"n-6fa6977267f2e73d","node_b_id":null,"fingerprint":null,"oeis_id":"A391403","in_catalog":1,"target":"reversed_triangle","row_alignment":"node_lengths","rows_matched":14,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":14,"target":"reversed_triangle","terms_compared":56},"in_catalog":true,"oeis_id":"A391403","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-626a43e248f57c55","kind":"oeis_match","node_a_id":"n-6fa6977267f2e73d","node_b_id":null,"fingerprint":null,"oeis_id":"A105868","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":12,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":12,"target":"triangle","terms_compared":80},"in_catalog":true,"oeis_id":"A105868","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-7d204d47ad61306c","kind":"oeis_match","node_a_id":"n-6fa6977267f2e73d","node_b_id":null,"fingerprint":null,"oeis_id":"A089627","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":15,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":15,"target":"reversed_triangle","terms_compared":120},"in_catalog":true,"oeis_id":"A089627","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f6819939f69b4957","kind":"oeis_match","node_a_id":"n-6fa6977267f2e73d","node_b_id":null,"fingerprint":null,"oeis_id":"A002426","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":"A002426","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A105868","relation":"identification"},{"oeis_id":"A089627","relation":"match_reversed_triangle"},{"oeis_id":"A391403","relation":"match_reversed_triangle"},{"oeis_id":"A002426","relation":"match_row_sums"},{"oeis_id":"A105868","relation":"match_triangle"}],"bfiles":[{"node_id":"n-6fa6977267f2e73d","oeis_id":"A002426","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":10239},{"node_id":"n-6fa6977267f2e73d","oeis_id":"A089627","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":120,"oeis_terms_agreeing":120,"oeis_terms_available":120,"term_count":990,"raw_size":10067},{"node_id":"n-6fa6977267f2e73d","oeis_id":"A105868","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":80,"oeis_terms_agreeing":80,"oeis_terms_available":80,"term_count":990,"raw_size":10067},{"node_id":"n-6fa6977267f2e73d","oeis_id":"A391403","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":56,"oeis_terms_agreeing":56,"oeis_terms_available":56,"term_count":992,"raw_size":18365}],"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: 0, 1;\n  2: 0, 2, 1;\n  3: 0, 0, 6,  1;\n  4: 0, 0, 6, 12,   1;\n  5: 0, 0, 0, 30,  20,   1;\n  6: 0, 0, 0, 20,  90,  30,    1;\n  7: 0, 0, 0,  0, 140, 210,   42,   1;\n  8: 0, 0, 0,  0,  70, 560,  420,  56,  1;\n  9: 0, 0, 0,  0,   0, 630, 1680, 756, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(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":"A089627","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,  0;\n  2: 1,  2,   0;\n  3: 1,  6,   0,    0;\n  4: 1, 12,   6,    0,   0;\n  5: 1, 20,  30,    0,   0, 0;\n  6: 1, 30,  90,   20,   0, 0, 0;\n  7: 1, 42, 210,  140,   0, 0, 0, 0;\n  8: 1, 56, 420,  560,  70, 0, 0, 0, 0;\n  9: 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n)]];\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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n)); 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":"A105868","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: 0, 1;\n  2: 0, 2, 1;\n  3: 0, 0, 6,  1;\n  4: 0, 0, 6, 12,   1;\n  5: 0, 0, 0, 30,  20,   1;\n  6: 0, 0, 0, 20,  90,  30,    1;\n  7: 0, 0, 0,  0, 140, 210,   42,   1;\n  8: 0, 0, 0,  0,  70, 560,  420,  56,  1;\n  9: 0, 0, 0,  0,   0, 630, 1680, 756, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n)]];\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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n)); 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":"A391403","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,  2;\n  3: 1,  6;\n  4: 1, 12,   6;\n  5: 1, 20,  30;\n  6: 1, 30,  90,   20;\n  7: 1, 42, 210,  140;\n  8: 1, 56, 420,  560,  70;\n  9: 1, 72, 756, 1680, 630;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = (2*n-1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n)]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}}],"derivation":{"root":"n-6fa6977267f2e73d","steps":[],"edges":[],"text":"A105868"},"headline":"A105868: Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).","chain":null}