A125695
Expansion of (sqrt(1+2x+9x^2)+x-1)/(2x).
Original entry on oeis.org
1, 2, -2, -2, 10, -6, -42, 102, 82, -782, 814, 3854, -12454, -5014, 98694, -142218, -472158, 1932258, 19038, -14816994, 27370410, 64159962, -334154442, 121279878, 2418497010, -5523511086, -8914677362, 61259567662, -44249714438
Offset: 0
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CoefficientList[Series[(Sqrt[1+2x+9x^2]+x-1)/(2x),{x,0,30}],x] (* Harvey P. Dale, Jul 25 2013 *)
A125693
Riordan array ((1-x)/(1-3*x), x*(1-x)/(1-3*x)).
Original entry on oeis.org
1, 2, 1, 6, 4, 1, 18, 16, 6, 1, 54, 60, 30, 8, 1, 162, 216, 134, 48, 10, 1, 486, 756, 558, 248, 70, 12, 1, 1458, 2592, 2214, 1168, 410, 96, 14, 1, 4374, 8748, 8478, 5160, 2150, 628, 126, 16, 1, 13122, 29160, 31590, 21744, 10442, 3624, 910, 160, 18, 1
Offset: 0
Triangle begins
1;
2, 1;
6, 4, 1;
18, 16, 6, 1;
54, 60, 30, 8, 1;
162, 216, 134, 48, 10, 1;
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Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j->
(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j) )))); # G. C. Greubel, Oct 28 2019
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T:= func< n,k | &+[(-1)^j*3^(n-k-j)*Binomial(k+1,j)*Binomial(n-j, n-k-j): j in [0..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 28 2019
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seq(seq( add( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j), j=0..n), k=0..n), n=0..10); # G. C. Greubel, Oct 28 2019
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T[0, 0]=1; T[1, 0]=2; T[1, 1]=1; T[n_, k_]/; 0<=k<=n:= T[n, k]= 3T[n-1, k] + T[n-1, k-1] - T[n-2, k-1]; T[, ]=0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
T[n_, k_]:= Sum[(-1)^j*3^(n-k-j)*Binomial[k+1,j]*Binomial[n-j,n-k-j], {j, 0, n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 28 2019 *)
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T(n,k) = sum(j=0,n, (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 28 2019
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[[sum( (-1)^j*3^(n-k-j)*binomial(k+1,j)*binomial(n-j, n-k-j) for j in (0..n) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 28 2019
A183875
Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)*x^n), where o.g.f. A(x) satisfies A(x)=(a+b*x*A(x))/(c-d*x*A(x)), a=1,b=2,c=1,d=2.
Original entry on oeis.org
1, 4, 1, 24, 8, 1, 176, 64, 12, 1, 1440, 544, 120, 16, 1, 12608, 4864, 1168, 192, 20, 1, 115584, 45184, 11424, 2112, 280, 24, 1, 1095424, 432128, 113088, 22528, 3440, 384, 28, 1, 10646016, 4227584, 1133952, 237824, 39840, 5216, 504, 32, 1, 105522176, 42115072, 11506944, 2505728, 448064, 65280, 7504, 640, 36, 1
Offset: 1
1,
4,1,
24,8,1,
176,64,12,1,
1440,544,120,16,1,
12608,4864,1168,192,20,1,
115584,45184,11424,2112,280,24,1,
1095424,432128,113088,22528,3440,384,28,1,
10646016,4227584,1133952,237824,39840,5216,504,32,1,
105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1
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T[n_, k_, a_, b_, c_, d_] := k/n Sum[Binomial[n, n - k - i] a^(k + i) b^(n - k - i) Binomial[i + n - 1, n - 1] c^(-i - n) d^i, {i, 0, n - k}];
T[n_, k_] := T[n, k, 1, 2, 1, 2];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, from formula *)
Showing 1-3 of 3 results.
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