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0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"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:        2,           1;\n  2:       11,          12,            1;\n  3:       84,         165,           44,            1;\n  4:      669,        2888,         1476,          126,           1;\n  5:     5350,       57313,        50612,         9446,         314,          1;\n  6:    42799,     1187532,      1869351,       611188,       48285,        720,         1;\n  7:   342392,    24895381,     72741360,     38698103,     5649232,     212055,      1568,       1;\n  8:  2739137,   523145400,   2899774642,   2488260890,   592107390,   42895372,    838766,    3306,    1;\n  9: 21913098, 10994270817, 116533305032, 162678522604, 59708846880, 7221572330, 281683024, 3082056, 6830, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-8+8*x)*P(n,x) = -(8*n-56+(2*n^2-31*n+183)*x+(n^3-20*n^2+123*n-248)*x^2+(12*n^2-136*n+156)*x^3+(32*n-32)*x^4)*P(n-2,x) - ((13*n-91)*x+(10*n^2-151*n+671)*x^2-(n^2-123*n+900)*x^3-(12*n-316)*x^4-32*x^5)*d/dx P(n-2,x) - ((3*n-21)*x^2+(3*n^2-45*n+192)*x^3+(36*n-276)*x^4+96*x^5)*d^2/dx^2 P(n-2,x) + (9*n-71+(2*n^2-31*n+192)*x+(20*n-156)*x^2+32*x^3)*P(n-1,x) + ((13*n-104)*x-(n-112)*x^2-8*x^3)*d/dx P(n-1,x) + ((3*n-24)*x^2+24*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 56 + (2*n^2 - 31*n + 183)*x + (n^3 - 20*n^2 + 123*n - 248)*x^2 + (12*n^2 - 136*n + 156)*x^3 + (32*n - 32)*x^4)*P[n - 2, x] - ((13*n - 91)*x + (10*n^2 - 151*n + 671)*x^2 - (n^2 - 123*n + 900)*x^3 - (12*n - 316)*x^4 - 32*x^5)*D[P[n - 2, x], x] - ((3*n - 21)*x^2 + (3*n^2 - 45*n + 192)*x^3 + (36*n - 276)*x^4 + 96*x^5)*D[P[n - 2, x], {x, 2}] + (9*n - 71 + (2*n^2 - 31*n + 192)*x + (20*n - 156)*x^2 + 32*x^3)*P[n - 1, x] + ((13*n - 104)*x - (n - 112)*x^2 - 8*x^3)*D[P[n - 1, x], x] + ((3*n - 24)*x^2 + 24*x^3)*D[P[n - 1, x], {x, 2}])/(n - 8 + 8*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n - 56 + (2*n^2 - 31*n + 183)*x + (n^3 - 20*n^2 + 123*n - 248)*x^2 + (12*n^2 - 136*n + 156)*x^3 + (32*n - 32)*x^4)*v[i - 2] - ((13*n - 91)*x + (10*n^2 - 151*n + 671)*x^2 - (n^2 - 123*n + 900)*x^3 - (12*n - 316)*x^4 - 32*x^5)*deriv(v[i - 2], x) - ((3*n - 21)*x^2 + (3*n^2 - 45*n + 192)*x^3 + (36*n - 276)*x^4 + 96*x^5)*deriv(deriv(v[i - 2], x), x) + (9*n - 71 + (2*n^2 - 31*n + 192)*x + (20*n - 156)*x^2 + 32*x^3)*v[i - 1] + ((13*n - 104)*x - (n - 112)*x^2 - 8*x^3)*deriv(v[i - 1], x) + ((3*n - 24)*x^2 + 24*x^3)*deriv(deriv(v[i - 1], x), x))/(n - 8 + 8*x)); 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}}],"derivation":{"root":"n-56eee146d6109546","steps":["D+1"],"edges":["e-2fc1c6309a1075f3"],"text":"A166972 \u2192 D+1"},"headline":"A166972 \u2192 D+1","chain":null}