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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:    3,      6,       3;\n  2:    7,     32,      48,       28,         5;\n  3:   15,    122,     345,      460,       305,        90,          7;\n  4:   31,    412,    1878,     4224,      5280,      3756,       1442,        248,          9;\n  5:   63,   1318,    8967,    30216,     58870,     70644,      52878,      23912,       5931,        630,         11;\n  6:  127,   4104,   39924,   189396,    525495,    927744,    1082368,     840792,     426141,     133080,      22572,       1524,         13;\n  7:  255,  12594,  170805,  1098708,   4121655,   9965742,   16362549,   18700056,   14948613,    8243310,    3021447,     683220,      81861,      3570,        15;\n  8:  511,  38324,  713586,  6073912,  29804090,  94482924,  206000578,  319877120,  359861760,  294286540,  173218694,   71529816,   19740214,   3330068,    287430,     8176,     17;\n  9: 1023, 116030, 2937891, 32547376, 204256580, 824697528, 2300669140, 4626050096, 6871916106, 7635462340, 6360818882, 3944004240, 1786844644, 571918424, 122052660, 15668752, 986255, 18414, 19;\n","recurrence_sentence":"The row polynomials satisfy (1+(4*n-4)*x)*P(n,x) = -(2+(8*n+2)*x+(10*n-4)*x^2+(8*n^2-12*n-6)*x^3+(20*n^2-34*n+6)*x^4+(16*n^3-40*n^2+24*n)*x^5)*P(n-2,x) - (x+(4*n+1)*x^2+(6*n-3)*x^3+(8*n^2-12*n-1)*x^4-(6*n-2)*x^5-(8*n^2-8*n)*x^6)*d/dx P(n-2,x) + (3+(12*n-6)*x+(12*n-9)*x^2+(16*n^2-28*n+12)*x^3)*P(n-1,x) + (x+(4*n-4)*x^2-x^3-(4*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*P[n - 2, x] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*D[P[n - 2, x], x] + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*P[n - 1, x] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2 + (8*n + 2)*x + (10*n - 4)*x^2 + (8*n^2 - 12*n - 6)*x^3 + (20*n^2 - 34*n + 6)*x^4 + (16*n^3 - 40*n^2 + 24*n)*x^5)*v[i - 2] - (x + (4*n + 1)*x^2 + (6*n - 3)*x^3 + (8*n^2 - 12*n - 1)*x^4 - (6*n - 2)*x^5 - (8*n^2 - 8*n)*x^6)*deriv(v[i - 2], x) + (3 + (12*n - 6)*x + (12*n - 9)*x^2 + (16*n^2 - 28*n + 12)*x^3)*v[i - 1] + (x + (4*n - 4)*x^2 - x^3 - (4*n - 4)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 4)*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":null,"headline":"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}