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begins:\n  0:  8;\n  1: 12,   18;\n  2: 16,   64,    32;\n  3: 20,  150,   200,     50;\n  4: 24,  288,   720,    480,     72;\n  5: 28,  490,  1960,   2450,    980,     98;\n  6: 32,  768,  4480,   8960,   6720,   1792,    128;\n  7: 36, 1134,  9072,  26460,  31752,  15876,   3024,   162;\n  8: 40, 1600, 16800,  67200, 117600,  94080,  33600,  4800,  200;\n  9: 44, 2178, 29040, 152460, 365904, 426888, 243936, 65340, 7260, 242;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2+9*n)*P(n,x) = (2*n^3+8*n^2+8*n-(4*n^3-5*n^2-26*n)*x)*P(n-1,x) - (4*n^3+5*n^2-7*n-2-(4*n^3+2*n^2+2*n-8)*x-(8*n^3-5*n^2-11*n+8)*x^2)*P(n-2,x) + (2*n^3-8*n-(8*n^3-3*n^2-26*n)*x+(10*n^3-6*n^2-28*n)*x^2-(4*n^3-3*n^2-10*n)*x^3)*P(n-3,x) for n >= 3, with P(0,x) = 8, P(1,x) = 12 + 18*x, P(2,x) = 16 + 64*x + 32*x^2.","mathematica":"P[0, x_] := 8;\nP[1, x_] := 12 + 18*x;\nP[2, x_] := 16 + 64*x + 32*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 8*n^2 + 8*n - (4*n^3 - 5*n^2 - 26*n)*x)*P[n - 1, x] - (4*n^3 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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A352880","target":"triangle","first_row":0,"oeis_offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:  8;\n   3: 12,   18;\n   4: 16,   64,    32;\n   5: 20,  150,   200,     50;\n   6: 24,  288,   720,    480,     72;\n   7: 28,  490,  1960,   2450,    980,     98;\n   8: 32,  768,  4480,   8960,   6720,   1792,    128;\n   9: 36, 1134,  9072,  26460,  31752,  15876,   3024,   162;\n  10: 40, 1600, 16800,  67200, 117600,  94080,  33600,  4800,  200;\n  11: 44, 2178, 29040, 152460, 365904, 426888, 243936, 65340, 7260, 242;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2-3*n-6)*P(n,x) = (2*n^3-4*n^2-(4*n^3-29*n^2+42*n)*x)*P(n-1,x) - (4*n^3-19*n^2+21*n-(4*n^3-22*n^2+42*n-36)*x-(8*n^3-53*n^2+105*n-54)*x^2)*P(n-2,x) + (2*n^3-12*n^2+16*n-(8*n^3-51*n^2+82*n-24)*x+(10*n^3-66*n^2+116*n-48)*x^2-(4*n^3-27*n^2+50*n-24)*x^3)*P(n-3,x) for n >= 5, with P(2,x) = 8, P(3,x) = 12 + 18*x, P(4,x) = 16 + 64*x + 32*x^2.","mathematica":"P[2, x_] := 8;\nP[3, x_] := 12 + 18*x;\nP[4, x_] := 16 + 64*x + 32*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 4*n^2 - (4*n^3 - 29*n^2 + 42*n)*x)*P[n - 1, x] - (4*n^3 - 19*n^2 + 21*n - (4*n^3 - 22*n^2 + 42*n - 36)*x - (8*n^3 - 53*n^2 + 105*n - 54)*x^2)*P[n - 2, x] + (2*n^3 - 12*n^2 + 16*n - (8*n^3 - 51*n^2 + 82*n - 24)*x + (10*n^3 - 66*n^2 + 116*n - 48)*x^2 - (4*n^3 - 27*n^2 + 50*n - 24)*x^3)*P[n - 3, x])/(3*n^2 - 3*n - 6)]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 1], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = 8; v[2] = 12 + 18*x; v[3] = 16 + 64*x + 32*x^2; for(n = 5, N, my(i = n - 1); v[i] = ((2*n^3 - 4*n^2 - (4*n^3 - 29*n^2 + 42*n)*x)*v[i - 1] - (4*n^3 - 19*n^2 + 21*n - (4*n^3 - 22*n^2 + 42*n - 36)*x - (8*n^3 - 53*n^2 + 105*n - 54)*x^2)*v[i - 2] + (2*n^3 - 12*n^2 + 16*n - (8*n^3 - 51*n^2 + 82*n - 24)*x + (10*n^3 - 66*n^2 + 116*n - 48)*x^2 - (4*n^3 - 27*n^2 + 50*n - 24)*x^3)*v[i - 3])/(3*n^2 - 3*n - 6)); 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(11)","reading":{"first_row":0,"offset":2,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u2299 C(n+1,k+1) (shift +2)"],"edges":["e-2d36970d4b706ae4"],"text":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2)"},"headline":"A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","chain":"A007318 \u2192 \u2299 C(n+1,k+1) (shift +2)"}