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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:    2,      1;\n  1:    4,      9,       6,        1;\n  2:    8,     39,      64,       44,        12,         1;\n  3:   16,    137,     406,      575,       420,       151,         22,          1;\n  4:   32,    443,    2084,     4850,      6336,      4812,       2068,        454,         40,          1;\n  5:   64,   1381,    9626,    33205,     66424,     82418,      64652,      31466,       8920,       1289,         74,          1;\n  6:  128,   4231,   41976,   202704,    572844,   1032843,    1236992,     995416,     531240,     180429,      35880,       3576,        140,         1;\n  7:  256,  12849,  177102,  1155643,   4396332,  10790073,   18023506,   21037563,   17286120,    9904267,    3845778,     957897,     138796,      9867,       270,        1;\n  8:  512,  38835,  732748,  6311774,  31322568, 100443742,  221747732,  349305774,  399846400,  334271180,  202647348,   87276970,   25701032,   4848546,    525292,    27338,     528,     1;\n  9: 1024, 117053, 2995906, 33526673, 212393424, 865548844, 2438118728, 4954717116, 7450172368, 8399008574, 7124365116, 4522260502, 2115511664, 709368012, 162903976, 23805596, 1965552, 76429, 1042, 1;\n","recurrence_sentence":"The row polynomials satisfy (12*n-12)*P(n,x) = -(48*n-(12*n-36)*x-(48*n^2-52*n+25)*x^2-(64*n-55)*x^3+(48*n^3-168*n^2+178*n-58)*x^4+(8*n^2-16*n+8)*x^5)*P(n-2,x) + (12*n*x+(24*n+9)*x^2-(32*n-2)*x^3+(10*n-58)*x^4+(40*n^2-96*n+47)*x^5+(4*n-4)*x^6)*d/dx P(n-2,x) + (12*n*x^2+9*x^3-(4*n+8)*x^4-9*x^5-(8*n-8)*x^6)*d^2/dx^2 P(n-2,x) + (48*n-24+6*x+(48*n^2-76*n+19)*x^2+(4*n-4)*x^3)*P(n-1,x) - (12*x+9*x^2+(20*n-20)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + x, P(1,x) = 4 + 9*x + 6*x^2 + x^3.","mathematica":"P[0, x_] := 2 + x;\nP[1, x_] := 4 + 9*x + 6*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n - (12*n - 36)*x - (48*n^2 - 52*n + 25)*x^2 - (64*n - 55)*x^3 + (48*n^3 - 168*n^2 + 178*n - 58)*x^4 + (8*n^2 - 16*n + 8)*x^5)*P[n - 2, x] + (12*n*x + (24*n + 9)*x^2 - (32*n - 2)*x^3 + (10*n - 58)*x^4 + (40*n^2 - 96*n + 47)*x^5 + (4*n - 4)*x^6)*D[P[n - 2, x], x] + (12*n*x^2 + 9*x^3 - (4*n + 8)*x^4 - 9*x^5 - (8*n - 8)*x^6)*D[P[n - 2, x], {x, 2}] + (48*n - 24 + 6*x + (48*n^2 - 76*n + 19)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - (12*x + 9*x^2 + (20*n - 20)*x^3)*D[P[n - 1, x], x])/(12*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; v[2] = 4 + 9*x + 6*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n - (12*n - 36)*x - (48*n^2 - 52*n + 25)*x^2 - (64*n - 55)*x^3 + (48*n^3 - 168*n^2 + 178*n - 58)*x^4 + (8*n^2 - 16*n + 8)*x^5)*v[i - 2] + (12*n*x + (24*n + 9)*x^2 - (32*n - 2)*x^3 + (10*n - 58)*x^4 + (40*n^2 - 96*n + 47)*x^5 + (4*n - 4)*x^6)*deriv(v[i - 2], x) + (12*n*x^2 + 9*x^3 - (4*n + 8)*x^4 - 9*x^5 - (8*n - 8)*x^6)*deriv(deriv(v[i - 2], x), x) + (48*n - 24 + 6*x + (48*n^2 - 76*n + 19)*x^2 + (4*n - 4)*x^3)*v[i - 1] - (12*x + 9*x^2 + (20*n - 20)*x^3)*deriv(v[i - 1], x))/(12*n - 12)); 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":null,"headline":"one_plus_derivative of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","chain":null}