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0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        6,         4;\n   3:       25,        52,         16;\n   4:      116,       452,        368,          64;\n   5:      641,      4000,       7088,        2816,         256;\n   6:     4170,     37572,     114976,      128960,       29184,         1024;\n   7:    31209,    376260,    1703936,     3483840,     2898944,       455680,         4096;\n   8:   263800,   4030292,   24583728,    75157760,   116717824,     78430208,      9859072,       16384;\n   9:  2484545,  46259424,  357943776,  1475783168,  3434121984,   4333297664,   2463293440,   263979008,      65536;\n  10: 25803886, 568896708, 5361331008, 28017332608, 87834886656, 165911006208, 177146937344, 87636934656, 8239054848, 262144;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16)*P(n,x) = -(9*n^3-80*n^2+255*n-288+(112*n^3-1096*n^2+3568*n-3864)*x+(480*n^3-5088*n^2+16800*n-17568)*x^2+(768*n^3-8576*n^2+28416*n-28800)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*P(n-3,x) - (n^3-8*n^2+23*n-24+(16*n^3-136*n^2+400*n-408)*x+(96*n^3-864*n^2+2592*n-2592)*x^2+(256*n^3-2432*n^2+7424*n-7296)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*d/dx P(n-3,x) + (n^3-30*n^2+117*n-128+(12*n^3-276*n^2+1024*n-1032)*x+(48*n^3-768*n^2+2480*n-2272)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*P(n-2,x) + (n^3-6*n^2+13*n-8+(12*n^3-84*n^2+192*n-136)*x+(48*n^3-384*n^2+944*n-736)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*d/dx P(n-2,x) + (15*n^2-43*n+32+(56*n^2-208*n+184)*x-(16*n^2-112*n+160)*x^2)*P(n-1,x) - (n^2-5*n+8+(8*n^2-48*n+72)*x+(16*n^2-112*n+160)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 6 + 4*x, P(3,x) = 25 + 52*x + 16*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 6 + 4*x;\nP[3, x_] := 25 + 52*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(9*n^3 - 80*n^2 + 255*n - 288 + (112*n^3 - 1096*n^2 + 3568*n - 3864)*x + (480*n^3 - 5088*n^2 + 16800*n - 17568)*x^2 + (768*n^3 - 8576*n^2 + 28416*n - 28800)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*D[P[n - 3, x], x] + (n^3 - 30*n^2 + 117*n - 128 + (12*n^3 - 276*n^2 + 1024*n - 1032)*x + (48*n^3 - 768*n^2 + 2480*n - 2272)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*D[P[n - 2, x], x] + (15*n^2 - 43*n + 32 + (56*n^2 - 208*n + 184)*x - (16*n^2 - 112*n + 160)*x^2)*P[n - 1, x] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; v[3] = 25 + 52*x + 16*x^2; for(n = 4, N, my(i = n); v[i] = (-(9*n^3 - 80*n^2 + 255*n - 288 + (112*n^3 - 1096*n^2 + 3568*n - 3864)*x + (480*n^3 - 5088*n^2 + 16800*n - 17568)*x^2 + (768*n^3 - 8576*n^2 + 28416*n - 28800)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*deriv(v[i - 3], x) + (n^3 - 30*n^2 + 117*n - 128 + (12*n^3 - 276*n^2 + 1024*n - 1032)*x + (48*n^3 - 768*n^2 + 2480*n - 2272)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*deriv(v[i - 2], x) + (15*n^2 - 43*n + 32 + (56*n^2 - 208*n + 184)*x - (16*n^2 - 112*n + 160)*x^2)*v[i - 1] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*deriv(v[i - 1], x))/(8*n - 16)); 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}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013611","D+1"],"edges":["e-5274dd88a300ad29","e-58cbd71ae1c9b507"],"text":"A046716 \u2192 basis A013611 \u2192 D+1"},"headline":"A046716 \u2192 basis A013611 \u2192 D+1","chain":null}