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Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["4","32","64"]],"card_rows":[["1"],["2","8"],["4","32","64"],["8","96","384","512"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611"],"edges":["e-d7d2422771947afc"],"text":"A130123 \u2192 basis A013611"},"headline":"A130123 \u2192 basis A013611","chain":null}},{"id":"l-fb3786c34a02a025","kind":"fingerprint_related","node_a_id":"n-8d4f9cf28f37366a","node_b_id":"n-7a0aecafcb4a073f","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-7a0aecafcb4a073f","other":{"id":"n-7a0aecafcb4a073f","name":"basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-4"],["1","8","16"]],"card_rows":[["1"],["-1","-4"],["1","8","16"],["-1","-12","-48","-64"]],"row_sums_preview":["1","-5","25","-125","625","-3125","15625","-78125"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013611"],"edges":["e-f6ed0d45ba75837f"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013611","chain":null}},{"id":"l-42272909a7562531","kind":"oeis_match","node_a_id":"n-8d4f9cf28f37366a","node_b_id":null,"fingerprint":null,"oeis_id":"A212699","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A212699","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A212699","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-8d4f9cf28f37366a","oeis_id":"A212699","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":15397}],"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:  0;\n  1:  4;\n  2:  8,   32;\n  3: 12,   96,   192;\n  4: 16,  192,   768,   1024;\n  5: 20,  320,  1920,   5120,   5120;\n  6: 24,  480,  3840,  15360,  30720,   24576;\n  7: 28,  672,  6720,  35840, 107520,  172032,  114688;\n  8: 32,  896, 10752,  71680, 286720,  688128,  917504,  524288;\n  9: 36, 1152, 16128, 129024, 645120, 2064384, 4128768, 4718592, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n + 4*n*x)*v[i - 1])/(n - 1)); 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-2e0409c0014ec240","steps":["D"],"edges":["e-b04b02754d5cebbf"],"text":"A013611 \u2192 D"},"headline":"A013611 \u2192 D","chain":null}