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Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-833b492313a93e6e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","32"],["100","640","1024"]],"card_rows":[["1"],["10","32"],["100","640","1024"],["1000","9600","30720","32768"]],"row_sums_preview":["1","42","1764","74088","3111696","130691232","5489031744","230539333248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","basis A013611"],"edges":["e-d7d2422771947afc","e-3d032f99625b4202"],"text":"A130123 \u2192 basis A013611 \u2192 basis A013611"},"headline":"A130123 \u2192 basis A013611 \u2192 basis A013611","chain":null}},{"id":"l-c6527c44edc30edb","kind":"fingerprint_related","node_a_id":"n-8bfae0a3670d90bd","node_b_id":"n-b451fc07991937e5","fingerprint":"f6-833b492313a93e6e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-833b492313a93e6e"},"other_id":"n-b451fc07991937e5","other":{"id":"n-b451fc07991937e5","name":"basis_map_A013611 of derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-833b492313a93e6e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["40","128"]],"card_rows":[["4"],["40","128"],["300","1920","3072"],["2000","19200","61440","65536"]],"row_sums_preview":["0","4","168","5292","148176","3889620","98018424","2401451388"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D","basis A013611"],"edges":["e-b04b02754d5cebbf","e-65067d14cdde68ad"],"text":"A013611 \u2192 D \u2192 basis A013611"},"headline":"A013611 \u2192 D \u2192 basis A013611","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:       25,        160,         256;\n  1:      125,       1200,        3840,         4096;\n  2:      625,       8000,       38400,        81920,         65536;\n  3:     3125,      50000,      320000,      1024000,       1638400,       1048576;\n  4:    15625,     300000,     2400000,     10240000,      24576000,      31457280,       16777216;\n  5:    78125,    1750000,    16800000,     89600000,     286720000,     550502400,      587202560,      268435456;\n  6:   390625,   10000000,   112000000,    716800000,    2867200000,    7340032000,    11744051200,    10737418240,     4294967296;\n  7:  1953125,   56250000,   720000000,   5376000000,   25804800000,   82575360000,   176160768000,   241591910400,   193273528320,    68719476736;\n  8:  9765625,  312500000,  4500000000,  38400000000,  215040000000,  825753600000,  2202009600000,  4026531840000,  4831838208000,  3435973836800,  1099511627776;\n  9: 48828125, 1718750000, 27500000000, 264000000000, 1689600000000, 7569408000000, 24222105600000, 55364812800000, 88583700480000, 94489280512000, 60473139527680, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 160*x + 256*x^2.","mathematica":"P[0, x_] := 25 + 160*x + 256*x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 25 + 160*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 16*x)*v[i - 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-2bc1923905556bf6","steps":["x \u21a6 x+1","reverse"],"edges":["e-2406b2ec50933490","e-f2ab5e4a2a83cb67"],"text":"A038319 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038319 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}