A277405
a(n) equals the constant term in the sum of all permutations of compositions of functions (1 + k*x) for k=1..n.
Original entry on oeis.org
1, 5, 40, 524, 10776, 327732, 13920096, 788050944, 57348311040, 5215111879680, 579420628853760, 77220215372770560, 12157472554474222080, 2232192933566250681600, 472721150641130889523200, 114371049117960857921126400, 31350167446592485414541721600, 9664519305841281076219121664000, 3328880054333616589332111409152000, 1273663039174670323519439513960448000
Offset: 1
Illustration of initial terms.
a(1) = 1, the constant term of (1+x);
a(2) = 5, the constant term of the sum of permutations of compositions of functions (1+x) and (1+2*x): (1+x)o(1+2*x) + (1+2*x)o(1+x) = (2*x + 2) + (2*x + 3) = 4*x + 5.
a(3) = 40, the constant term of the sum of permutations of compositions of functions (1+x), (1+2*x), and (1+3*x): (1+x)o(1+2*x)o(1+3*x) + (1+x)o(1+3*x)o(1+2*x) + (1+2*x)o(1+1*x)o(1+3*x) + (1+2*x)o(1+3*x)o(1+1*x) + (1+3*x)o(1+1*x)o(1+2*x) + (1+3*x)o(1+2*x)o(1+1*x) = (6*x + 4) + (6*x + 5) + (6*x + 5) + (6*x + 9) + (6*x + 7) + (6*x + 10) = 36*x + 40.
etc.
Alternatively,
a(2) = 5 = Sum_{i=1..2} (1+i),
a(3) = 40 = Sum_{i=1..3, j=1..3, j<>i} (1 + i*(1+j)),
a(4) = 524 = Sum_{i=1..4, j=1..4, k=1..4, k<>j<>i, k<>i} (1 + i*(1 + j*(1+k))), etc.
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{a(n) = sum(k=0, n-1, k!*(n-k)! * sum(i=0, n-k+1, (-1)^(n-i+1) * stirling(i, n-k+1, 2) * stirling(n+1, i, 1)))}
for(n=1, 20, print1(a(n), ", "))
A277406
a(n) equals the sum of all permutations of compositions of functions (1 + k*x) for k=1..n, evaluated at x=1.
Original entry on oeis.org
1, 2, 9, 76, 1100, 25176, 846132, 39321696, 2413753344, 189030205440, 18383301319680, 2172771551093760, 306662748175330560, 50933260598106862080, 9832247390118248121600, 2182733403365330313523200, 552134185815355910465126400, 157863713952139655599757721600, 50654908373638564216229105664000
Offset: 0
Illustration of initial terms.
a(0) = 1, by convention;
a(1) = 2, the function (1+x) evaluated at x=1;
a(2) = 9, the sum of permutations of compositions of functions (1+x) and (1+2*x), evaluated at x=1: (1+x)o(1+2*x) + (1+2*x)o(1+x) = (2*x + 2) + (2*x + 3) = 4*x + 5.
a(3) = 76, the sum of permutations of compositions of functions (1+x), (1+2*x), and (1+3*x), evaluated at x=1: (1+x)o(1+2*x)o(1+3*x) + (1+x)o(1+3*x)o(1+2*x) + (1+2*x)o(1+1*x)o(1+3*x) + (1+2*x)o(1+3*x)o(1+1*x) + (1+3*x)o(1+1*x)o(1+2*x) + (1+3*x)o(1+2*x)o(1+1*x) = (6*x + 4) + (6*x + 5) + (6*x + 5) + (6*x + 9) + (6*x + 7) + (6*x + 10) = 36*x + 40.
etc.
Alternatively,
a(1) = 2 = Sum_{i=1..1} (1+i),
a(2) = 9 = Sum_{i=1..2, j=1..2, j<>i} (1 + i*(1+j)),
a(3) = 76 = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*(1 + j*(1+k))),
a(4) = 1100 = Sum_{i=1..4, j=1..4, k=1..4, m=1..4, i,j,k,m distinct} (1 + i*(1 + j*(1 + k*(1+m)))), etc.
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Table[Sum[k!*(n-k)! * Sum[(-1)^(n-i+1) * StirlingS2[i, n-k+1] * StirlingS1[n+1, i], {i, 0, n-k+1}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 27 2016 *)
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{a(n) = sum(k=0,n, k!*(n-k)! * sum(i=0, n-k+1, (-1)^(n-i+1) * stirling(i, n-k+1, 2) * stirling(n+1, i, 1)))}
for(n=0,20,print1(a(n),", "))
A277408
Triangle, read by rows, where the g.f. of row n equals the sum of permutations of compositions of functions (1 + k*y*x) for k=1..n with parameter y independent of variable x, as evaluated at x=1.
Original entry on oeis.org
1, 1, 1, 2, 3, 4, 6, 12, 22, 36, 24, 60, 140, 300, 576, 120, 360, 1020, 2700, 6576, 14400, 720, 2520, 8400, 26460, 77952, 211680, 518400, 5040, 20160, 77280, 282240, 974736, 3151680, 9408960, 25401600, 40320, 181440, 786240, 3265920, 12930624, 48444480, 170098560, 552303360, 1625702400, 362880, 1814400, 8769600, 40824000, 182226240, 775656000, 3126297600, 11820816000, 41391544320, 131681894400, 3628800, 19958400, 106444800, 548856000, 2726317440, 12989592000, 59044550400, 254303280000, 1028448368640, 3856920883200, 13168189440000, 39916800, 239500800, 1397088000, 7903526400, 43233886080, 227885011200, 1152535824000, 5563643500800, 25464033745920, 109530230261760, 437429486592000, 1593350922240000
Offset: 0
Illustration of initial row polynomials.
R_0(y) = 1;
R_1(y) = 1 + y;
R_2(y) = 2 + 3*y + 4*y^2;
R_3(y) = 6 + 12*y + 22*y^2 + 36*y^3;
R_4(y) = 24 + 60*y + 140*y^2 + 300*y^3 + 576*y^4;
R_5(y) = 120 + 360*y + 1020*y^2 + 2700*y^3 + 6576*y^4 + 14400*y^5;
R_6(y) = 720 + 2520*y + 8400*y^2 + 26460*y^3 + 77952*y^4 + 211680*y^5 + 518400*y^6;
R_7(y) = 5040 + 20160*y + 77280*y^2 + 282240*y^3 + 974736*y^4 + 3151680*y^5 + 9408960*y^6 + 25401600*y^7;
...
Generating method.
R_0(y) = 1, by convention;
R_1(y) = Sum_{i=1..1} (1 + i*y);
R_2(y) = Sum_{i=1..2, j=1..2, j<>i} (1 + i*y*(1 + j*y));
R_3(y) = Sum_{i=1..3, j=1..3, k=1..3, i,j,k distinct} (1 + i*y*(1 + j*y*(1 + k*y)));
R_4(y) = Sum_{i=1..4, j=1..4, k=1..4, m=1..4, i,j,k,m distinct} (1 + i*y*(1 + j*y*(1 + k*y*(1 + m*y))));
etc.
This triangle of coefficients begins:
1;
1, 1;
2, 3, 4;
6, 12, 22, 36;
24, 60, 140, 300, 576;
120, 360, 1020, 2700, 6576, 14400;
720, 2520, 8400, 26460, 77952, 211680, 518400;
5040, 20160, 77280, 282240, 974736, 3151680, 9408960, 25401600;
40320, 181440, 786240, 3265920, 12930624, 48444480, 170098560, 552303360, 1625702400; ...
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{T(n,k) = k!*(n-k)! * sum(i=0,n-k+1, (-1)^(n-i+1) * stirling(i,n-k+1,2) * stirling(n+1,i,1))}
for(n=0,11,for(k=0,n,print1( T(n,k) ,", "));print(""))
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{T(n, k) = if( k<0 || k>n, 0, n! * k! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^(n+1), k))}; /* Michael Somos, May 10 2017 */
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