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0","first_index":1,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-1"],["1","3","1"]],"card_rows":[["1"],["-1","-1"],["1","3","1"],["-1","-7","-6","-1"]],"row_sums_preview":["1","-2","5","-15","52","-203","877","-4140"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A008277"],"edges":["e-9c20f6f9d09a133d"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis 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begins:\n   1: 1;\n   2: 1,  1;\n   3: 1,  3,   1;\n   4: 1,  6,   7,    1;\n   5: 1, 10,  25,   15,     1;\n   6: 1, 15,  65,   90,    31,     1;\n   7: 1, 21, 140,  350,   301,    63,     1;\n   8: 1, 28, 266, 1050,  1701,   966,   127,    1;\n   9: 1, 36, 462, 2646,  6951,  7770,  3025,  255,   1;\n  10: 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A008277","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   1;\n   3: 1,   3,    1;\n   4: 1,   7,    6,     1;\n   5: 1,  15,   25,    10,     1;\n   6: 1,  31,   90,    65,    15,     1;\n   7: 1,  63,  301,   350,   140,    21,    1;\n   8: 1, 127,  966,  1701,  1050,   266,   28,   1;\n   9: 1, 255, 3025,  7770,  6951,  2646,  462,  36,  1;\n  10: 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A008278","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  1;\n   3: 1,  3,   1;\n   4: 1,  6,   7,    1;\n   5: 1, 10,  25,   15,     1;\n   6: 1, 15,  65,   90,    31,     1;\n   7: 1, 21, 140,  350,   301,    63,     1;\n   8: 1, 28, 266, 1050,  1701,   966,   127,    1;\n   9: 1, 36, 462, 2646,  6951,  7770,  3025,  255,   1;\n  10: 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-f277d949034445ba","steps":[],"edges":[],"text":"A008278"},"headline":"A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.","chain":null}