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T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","30","6"]],"card_rows":[["1"],["4","2"],["24","30","6"],["160","380","180","20"]],"row_sums_preview":["1","6","60","740","10570","169848","3015012","58368024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A008277"],"edges":["e-398f569c764d9a1d"],"text":"A356546 \u2192 basis A008277"},"headline":"A356546 \u2192 basis A008277","chain":null}},{"id":"l-d3ae85762ab3d850","kind":"fingerprint_related","node_a_id":"n-a514cae29d6b9cc2","node_b_id":"n-c072d889dfa1d499","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c072d889dfa1d499","other":{"id":"n-c072d889dfa1d499","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","6"],["84","105","21"]],"card_rows":[["1"],["12","6"],["84","105","21"],["448","1064","504","56"]],"row_sums_preview":["1","18","210","2072","19026","169848","1507506","13469544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A008277"],"edges":["e-af56d17dbcb6f76e"],"text":"A178822 \u2192 basis A008277"},"headline":"A178822 \u2192 basis A008277","chain":null}},{"id":"l-d6f7a81e5244d438","kind":"fingerprint_related","node_a_id":"n-a514cae29d6b9cc2","node_b_id":"n-c8044fd3d7ed3bae","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c8044fd3d7ed3bae","other":{"id":"n-c8044fd3d7ed3bae","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","5"],["60","75","15"]],"card_rows":[["1"],["10","5"],["60","75","15"],["280","665","315","35"]],"row_sums_preview":["1","15","150","1295","10570","84924","685230","5612310"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A008277"],"edges":["e-2482ec2370e1f9d0"],"text":"A178821 \u2192 basis A008277"},"headline":"A178821 \u2192 basis A008277","chain":null}}],"oeis":[],"bfiles":[],"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,     -1;\n  2:    4,      5,       1;\n  3:   -8,    -19,      -9,      -1;\n  4:   16,     65,      55,      14,       1;\n  5:  -32,   -211,    -285,    -125,     -20,     -1;\n  6:   64,    665,    1351,     910,     245,     27,      1;\n  7: -128,  -2059,   -6069,   -5901,   -2380,   -434,    -35,    -1;\n  8:  256,   6305,   26335,   35574,   20181,   5418,    714,    44,   1;\n  9: -512, -19171, -111645, -204205, -156660, -58107, -11130, -1110, -54, -1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+x)*P(n-1,x) - x*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[-(2 + x)*P[n - 1, x] - x*D[P[n - 1, x], x]];\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] = -(2 + x)*v[i - 1] - x*deriv(v[i - 1], x)); 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}}],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A008277"],"edges":["e-0bd7ea1430928905"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A008277","chain":null}