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A013609","chain":null}},{"id":"l-ff38278ef0fb7a46","kind":"fingerprint_related","node_a_id":"n-c7809537965f400d","node_b_id":"n-75a6475b3e851dc5","fingerprint":"f6-77a7ed3b1d5f257e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-77a7ed3b1d5f257e"},"other_id":"n-75a6475b3e851dc5","other":{"id":"n-75a6475b3e851dc5","name":"Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-77a7ed3b1d5f257e","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["6","12","6"]],"card_rows":[["1"],["2","2"],["6","12","6"],["24","72","72","24"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A156992"],"derivation":{"root":"n-75a6475b3e851dc5","steps":[],"edges":[],"text":"A156992"},"headline":"A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","chain":null}},{"id":"l-04eb9a55564cc7d6","kind":"oeis_match","node_a_id":"n-c7809537965f400d","node_b_id":null,"fingerprint":null,"oeis_id":"A001813","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":"A001813","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-379f00612869a13e","kind":"oeis_match","node_a_id":"n-c7809537965f400d","node_b_id":null,"fingerprint":null,"oeis_id":"A085881","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A085881","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a62aed65f630cbd9","kind":"oeis_match","node_a_id":"n-c7809537965f400d","node_b_id":null,"fingerprint":null,"oeis_id":"A085881","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A085881","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A085881","relation":"identification"},{"oeis_id":"A085881","relation":"match_reversed_triangle"},{"oeis_id":"A001813","relation":"match_row_sums"},{"oeis_id":"A085881","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c7809537965f400d","oeis_id":"A001813","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":45809},{"node_id":"n-c7809537965f400d","oeis_id":"A085881","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":49302},{"node_id":"n-c7809537965f400d","oeis_id":"A085881","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":49302}],"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:        1,         1;\n  2:        3,         6,          3;\n  3:       15,        45,         45,         15;\n  4:      105,       420,        630,        420,        105;\n  5:      945,      4725,       9450,       9450,       4725,        945;\n  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;\n  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;\n  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;\n  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*x)*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*n - 1 + (2*n - 1)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + (2*n - 1)*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}},{"oeis_id":"A085881","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:        1,         1;\n  2:        3,         6,          3;\n  3:       15,        45,         45,         15;\n  4:      105,       420,        630,        420,        105;\n  5:      945,      4725,       9450,       9450,       4725,        945;\n  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;\n  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;\n  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;\n  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*x)*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*n - 1 + (2*n - 1)*x)*P[n - 1, x]];\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 + (2*n - 1)*x)*v[i - 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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A085881","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,         1;\n  2:        3,         6,          3;\n  3:       15,        45,         45,         15;\n  4:      105,       420,        630,        420,        105;\n  5:      945,      4725,       9450,       9450,       4725,        945;\n  6:    10395,     62370,     155925,     207900,     155925,      62370,      10395;\n  7:   135135,    945945,    2837835,    4729725,    4729725,    2837835,     945945,     135135;\n  8:  2027025,  16216200,   56756700,  113513400,  141891750,  113513400,   56756700,   16216200,   2027025;\n  9: 34459425, 310134825, 1240539300, 2894591700, 4341887550, 4341887550, 2894591700, 1240539300, 310134825, 34459425;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2*n-1+(2*n-1)*x)*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*n - 1 + (2*n - 1)*x)*P[n - 1, x]];\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 + (2*n - 1)*x)*v[i - 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":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c7809537965f400d","steps":[],"edges":[],"text":"A085881"},"headline":"A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","chain":null}