A144254
Eigentriangle by rows, termwise products of A078812 and its eigensequence, A125274.
Original entry on oeis.org
1, 2, 1, 3, 4, 3, 4, 10, 18, 10, 5, 20, 63, 80, 42, 6, 35, 168, 360, 420, 210, 7, 56, 378, 1200, 2310, 2520, 1199, 8, 84, 756, 3300, 9240, 16380, 16786, 7670, 9, 120, 1386, 7920, 30030, 76440, 125895, 122720, 54224, 10, 165, 2376, 17160, 84084, 286650
Offset: 1
First few rows of the triangle =
1;
2, 1;
3, 4, 3;
4, 10, 18, 10;
5, 20, 63, 80, 42;
6, 35, 168, 360, 420, 210;
7, 56, 378, 1200, 2310, 2520, 1199;
...
Triangle A078812 begins:
1;
2, 1;
3, 4, 1;
4, 10, 6, 1;
5, 20, 21, 8, 1;
...
Its eigensequence = A125274: (1, 1, 3, 10, 42, 210, 1199,...).
Row 3 of triangle A144253 = termwise products of (4, 10, 6, 1) and (1, 1, 3, 10) = (4*1, 10*1, 6*3, 1*10).
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.
-
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}]
A351817
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x)^4.
Original entry on oeis.org
1, 1, 5, 23, 139, 1052, 9166, 90073, 989205, 11981051, 158149438, 2255926638, 34549223880, 564898101239, 9812669832553, 180324597042263, 3492960489714519, 71092066388237562, 1516044005669227542, 33788707128788508476, 785270646437483414261, 18992014442689191510460
Offset: 0
-
nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^4]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3 k + 2, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
A351818
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^5) / (1 - x)^5.
Original entry on oeis.org
1, 1, 6, 31, 211, 1841, 18547, 210664, 2682657, 37807531, 581985596, 9696297528, 173702897000, 3327063115248, 67790086866271, 1462900566163696, 33310115601839624, 797687851718024035, 20032231443590167914, 526189230537615409571, 14423255501358439152231
Offset: 0
-
nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^5]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 4 k + 3, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
A351659
G.f. A(x) satisfies: A(x) = 1 - x * A(x/(1 - x)^2) / (1 - x)^2.
Original entry on oeis.org
1, -1, -1, 2, 10, 10, -97, -638, -1316, 9908, 118713, 560533, -697429, -38229322, -364288567, -1441996161, 11586777849, 281338444108, 2772828770441, 10249821640498, -170439385810217, -4104012197171264, -46232949019802137, -204897893603728741, 3708422726478663919
Offset: 0
-
nmax = 24; A[] = 0; Do[A[x] = 1 - x A[x/(1 - x)^2]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n + k, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
Showing 1-6 of 6 results.
Comments