A125273
Eigensequence of triangle A085478: a(n) = Sum_{k=0..n-1} A085478(n-1,k)*a(k) for n > 0 with a(0) = 1.
Original entry on oeis.org
1, 1, 2, 6, 23, 106, 567, 3434, 23137, 171174, 1376525, 11934581, 110817423, 1095896195, 11487974708, 127137087319, 1480232557526, 18075052037054, 230855220112093, 3076513227516437, 42686898298650967, 615457369662333260
Offset: 0
a(3) = 1*(1) + 3*(1) + 1*(2) = 6;
a(4) = 1*(1) + 6*(1) + 5*(2) + 1*(6) = 23;
a(5) = 1*(1) + 10*(1) + 15*(2) + 7*(6) + 1*(23) = 106.
Triangle A085478(n,k) = binomial(n+k, n-k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 1;
1, 3, 1;
1, 6, 5, 1;
1, 10, 15, 7, 1;
1, 15, 35, 28, 9, 1;
...
where g.f. of column k = 1/(1-x)^(2*k+1).
- Seiichi Manyama, Table of n, a(n) for n = 0..517
- Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
- Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
- Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.
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A125273=ConstantArray[0,20]; A125273[[1]]=1; Do[A125273[[n]]=1+Sum[A125273[[k]]*Binomial[n+k-1, n-k-1],{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125273}] (* Vaclav Kotesovec, Dec 10 2013 *)
-
a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(n+k-1, n-k-1)))
A351816
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^3) / (1 - x)^3.
Original entry on oeis.org
1, 1, 4, 16, 83, 526, 3826, 31338, 285556, 2857831, 31083421, 364523891, 4579906098, 61313286380, 870531542926, 13055593578453, 206097824225131, 3414146518958089, 59189048364709453, 1071264611091540458, 20197719805598878119, 395917304689782855768
Offset: 0
-
nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^3]/(1 - x)^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 2 k + 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
A351814
G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x).
Original entry on oeis.org
1, 1, 2, 8, 42, 272, 2115, 19010, 192760, 2172468, 26896081, 362184998, 5262526484, 81969555736, 1361249430071, 23989460080079, 446832403813788, 8765575657218860, 180544405959236487, 3893718987163468969, 87711985393624557487, 2059264143275898894916
Offset: 0
-
nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^4]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
A351815
G.f. A(x) satisfies A(x) = 1 + x * A(x/(1 - x)^5) / (1 - x).
Original entry on oeis.org
1, 1, 2, 9, 53, 386, 3422, 35300, 412084, 5364255, 76952267, 1203835714, 20362911276, 369906504888, 7175947738672, 147944905766929, 3227970924123268, 74264452788294013, 1795825803391367571, 45514495928632484735, 1205981001167335524448, 33331235326744168532151
Offset: 0
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nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^5]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 4 k - 1, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Showing 1-4 of 4 results.