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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:  4,    5,     1;\n  2:  8,   33,    34,     11,      1;\n  3: 12,   93,   230,    227,     96,      17,       1;\n  4: 16,  185,   778,   1515,   1476,     741,     190,       23,        1;\n  5: 20,  309,  1870,   5731,   9704,    9417,    5314,     1735,      316,       29,        1;\n  6: 24,  465,  3698,  15755,  39564,   61117,   59270,    36375,    14080,     3369,      474,      35,       1;\n  7: 28,  653,  6454,  35491, 120176,  262849,  380554,   369287,   241348,   105973,    30878,    5803,     664,     41,      1;\n  8: 32,  873, 10330,  69867, 300116,  862165, 1703246,  2349975,  2283016,  1566593,   757890,  256387,   59580,   9197,    886,    47,    1;\n  9: 36, 1125, 15518, 124835, 652536, 2340345, 5937490, 10847175, 14419532, 14027213, 10006470, 5227115, 1988000, 543969, 104826, 13711, 1140, 53, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-20*n+12+(8*n^2-10*n-7)*x+(12*n-10)*x^2+(2*n+4)*x^3+x^4)*P(n,x) = -((8*n^2-4*n)*x^2+(8*n^2+6*n-9)*x^3+(12*n+2)*x^4+(2*n+6)*x^5+x^6)*P(n-2,x) + (8*n^2-12*n+(40*n^2-50*n-27)*x+(40*n^2-8*n-78)*x^2+(8*n^2+48*n-25)*x^3+(20*n+23)*x^4+(2*n+10)*x^5+x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*P[n - 2, x] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*P[n - 1, x])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 5*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 - 4*n)*x^2 + (8*n^2 + 6*n - 9)*x^3 + (12*n + 2)*x^4 + (2*n + 6)*x^5 + x^6)*v[i - 2] + (8*n^2 - 12*n + (40*n^2 - 50*n - 27)*x + (40*n^2 - 8*n - 78)*x^2 + (8*n^2 + 48*n - 25)*x^3 + (20*n + 23)*x^4 + (2*n + 10)*x^5 + x^6)*v[i - 1])/(8*n^2 - 20*n + 12 + (8*n^2 - 10*n - 7)*x + (12*n - 10)*x^2 + (2*n + 4)*x^3 + x^4)); 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-cdb0db4ad473ee28","steps":["basis A008288","D+1"],"edges":["e-054f97e0823e3ab6","e-634120a929c55ce7"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 D+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 D+1","chain":null}