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begins:\n  0:    3;\n  1:   15,    15;\n  2:   45,    90,    45;\n  3:  105,   315,   315,    105;\n  4:  210,   840,  1260,    840,    210;\n  5:  378,  1890,  3780,   3780,   1890,    378;\n  6:  630,  3780,  9450,  12600,   9450,   3780,    630;\n  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;\n  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;\n  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*x)*v[i - 1])/(n)); 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":"A344565","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:    3;\n  1:   15,    15;\n  2:   45,    90,    45;\n  3:  105,   315,   315,    105;\n  4:  210,   840,  1260,    840,    210;\n  5:  378,  1890,  3780,   3780,   1890,    378;\n  6:  630,  3780,  9450,  12600,   9450,   3780,    630;\n  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;\n  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;\n  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*x)*P[n - 1, x])/(n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*x)*v[i - 1])/(n)); 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":"A344565","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:    3;\n  1:   15,    15;\n  2:   45,    90,    45;\n  3:  105,   315,   315,    105;\n  4:  210,   840,  1260,    840,    210;\n  5:  378,  1890,  3780,   3780,   1890,    378;\n  6:  630,  3780,  9450,  12600,   9450,   3780,    630;\n  7:  990,  6930, 20790,  34650,  34650,  20790,   6930,   990;\n  8: 1485, 11880, 41580,  83160, 103950,  83160,  41580, 11880,  1485;\n  9: 2145, 19305, 77220, 180180, 270270, 270270, 180180, 77220, 19305, 2145;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (n+4+(n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n + 4)*x)*P[n - 1, x])/(n)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n + 4)*x)*v[i - 1])/(n)); 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-dd4aa31f3d11d152","steps":[],"edges":[],"text":"A344565"},"headline":"A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","chain":null}