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+ T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1."},{"id":"e-78c582ff5f7b6a70","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-4124f4242e7d4c67","conclusion_name":"Triangle read by rows, A007318 rows repeated."},{"id":"e-1714c7c9f2e3e607","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-67b0bcb95873e26d","conclusion_name":"mobius_x_over_x_plus_one of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1."},{"id":"e-218f1af72daab700","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-4db5974bca51a753","conclusion_name":"derivative of Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k 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begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  3,  1;\n  3: 1,  4,  4,   1;\n  4: 1,  6, 11,   6,   1;\n  5: 1,  7, 17,  17,   7,   1;\n  6: 1,  9, 30,  45,  30,   9,   1;\n  7: 1, 10, 39,  75,  75,  39,  10,  1;\n  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;\n  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2]); 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":"A026374","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: 1,  3,  1;\n  3: 1,  4,  4,   1;\n  4: 1,  6, 11,   6,   1;\n  5: 1,  7, 17,  17,   7,   1;\n  6: 1,  9, 30,  45,  30,   9,   1;\n  7: 1, 10, 39,  75,  75,  39,  10,  1;\n  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;\n  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2]); 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":"A026374","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: 1,  3,  1;\n  3: 1,  4,  4,   1;\n  4: 1,  6, 11,   6,   1;\n  5: 1,  7, 17,  17,   7,   1;\n  6: 1,  9, 30,  45,  30,   9,   1;\n  7: 1, 10, 39,  75,  75,  39,  10,  1;\n  8: 1, 12, 58, 144, 195, 144,  58, 12,  1;\n  9: 1, 13, 70, 202, 339, 339, 202, 70, 13, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+3*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, 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; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2]); 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-4e0b19699f4cf0db","steps":[],"edges":[],"text":"A026374"},"headline":"A026374: Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","chain":null}