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T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 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:      1,       5;\n  2:      2,      16,       23;\n  3:      6,      62,      160,       120;\n  4:     24,     296,     1059,      1486,       724;\n  5:    120,    1696,     7609,     15211,     14154,      5012;\n  6:    720,   11376,    60696,    155184,    208711,    142850,     39332;\n  7:   5040,   87552,   537660,   1659724,   2885936,   2880216,   1544624,    345832;\n  8:  40320,  760896,  5260428,  18918216,  40054845,  52035636,  40930188,  17942928,   3371976;\n  9: 362880, 7372224, 56468412, 230917412, 573522609, 912191313, 938318656, 605806740, 223750824, 36135792;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = -(n^2-2*n+1+(5*n^2-6*n+1)*x+(14*n^2-16*n-7)*x^2+(26*n^2-40*n-17)*x^3+(29*n^2-54*n-14)*x^4+(17*n^2-34*n-4)*x^5+(4*n^2-8*n)*x^6)*P(n-2,x) + ((4*n-4)*x^2+(20*n-16)*x^3+(40*n-24)*x^4+(40*n-16)*x^5+(20*n-4)*x^6+4*n*x^7)*d/dx P(n-2,x) + (2*n-1+8*n*x+(16*n+2)*x^2+(20*n-4)*x^3+(14*n-9)*x^4+(4*n-4)*x^5)*P(n-1,x) - (4*x^2+16*x^3+24*x^4+16*x^5+4*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 + (5*n^2 - 6*n + 1)*x + (14*n^2 - 16*n - 7)*x^2 + (26*n^2 - 40*n - 17)*x^3 + (29*n^2 - 54*n - 14)*x^4 + (17*n^2 - 34*n - 4)*x^5 + (4*n^2 - 8*n)*x^6)*P[n - 2, x] + ((4*n - 4)*x^2 + (20*n - 16)*x^3 + (40*n - 24)*x^4 + (40*n - 16)*x^5 + (20*n - 4)*x^6 + 4*n*x^7)*D[P[n - 2, x], x] + (2*n - 1 + 8*n*x + (16*n + 2)*x^2 + (20*n - 4)*x^3 + (14*n - 9)*x^4 + (4*n - 4)*x^5)*P[n - 1, x] - (4*x^2 + 16*x^3 + 24*x^4 + 16*x^5 + 4*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 + (5*n^2 - 6*n + 1)*x + (14*n^2 - 16*n - 7)*x^2 + (26*n^2 - 40*n - 17)*x^3 + (29*n^2 - 54*n - 14)*x^4 + (17*n^2 - 34*n - 4)*x^5 + (4*n^2 - 8*n)*x^6)*v[i - 2] + ((4*n - 4)*x^2 + (20*n - 16)*x^3 + (40*n - 24)*x^4 + (40*n - 16)*x^5 + (20*n - 4)*x^6 + 4*n*x^7)*deriv(v[i - 2], x) + (2*n - 1 + 8*n*x + (16*n + 2)*x^2 + (20*n - 4)*x^3 + (14*n - 9)*x^4 + (4*n - 4)*x^5)*v[i - 1] - (4*x^2 + 16*x^3 + 24*x^4 + 16*x^5 + 4*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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-3f4ccde57d57bff1","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-4e78d8cc10b1cff9","e-b47bf2d0ff6f14a7"],"text":"A028421 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A028421 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}