This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103371 #101 Jun 02 2025 16:49:53
%S A103371 1,2,1,3,6,1,4,18,12,1,5,40,60,20,1,6,75,200,150,30,1,7,126,525,700,
%T A103371 315,42,1,8,196,1176,2450,1960,588,56,1,9,288,2352,7056,8820,4704,
%U A103371 1008,72,1,10,405,4320,17640,31752,26460,10080,1620,90,1,11,550,7425,39600,97020
%N A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).
%C A103371 Columns include A000027, A002411, A004302, A108647, A134287. Row sums are C(2n+1,n+1) or A001700.
%C A103371 T(n-1,k-1) is the number of ways to put n identical objects into k of altogether n distinguishable boxes. See the partition array A035206 from which this triangle arises after summing over all entries related to partitions with fixed part number k.
%C A103371 T(n, k) is also the number of order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - _Abdullahi Umar_, Oct 02 2008
%C A103371 The o.g.f. of the (n+1)-th diagonal is given by G(n, x) = (n+1)*Sum_{k=1..n} A001263(n, k)*x^(k-1) / (1 - x)^(2*n+1), for n >= 1 and for n = 0 it is G(0, x) = 1/(1-x). - _Wolfdieter Lang_, Jul 31 2017
%H A103371 Reinhard Zumkeller, <a href="/A103371/b103371.txt">Rows n = 0..125 of table, flattened</a>
%H A103371 Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, <a href="https://arxiv.org/abs/2010.11157">Refined Catalan and Narayana cyclic sieving</a>, arXiv:2010.11157 [math.CO], 2020.
%H A103371 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.
%H A103371 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 16.
%H A103371 R. Cori and G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv:1306.4628 [math.CO], 2013.
%H A103371 Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H A103371 A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-005-0553-6">Combinatorial results for semigroups of order-preserving full transformations</a>, Semigroup Forum 72 (2006), 51-62.
%F A103371 Number triangle T(n, k) = C(n, n-k)*C(n+1, n-k) = C(n, k)*C(n+1, k+1); Column k of this triangle has g.f. Sum_{j=0..k} (C(k, j)*C(k+1, j) * x^(k+j))/(1-x)^(2*k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)*C(n+1, k).
%F A103371 T(n,k) = C(n, k)*Sum_{j=0..(n-k)} C(n-j, k). - _Paul Barry_, Jan 12 2006
%F A103371 T(n,k) = (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle (with offset [1,1]).
%F A103371 O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. of A001263, Narayana triangle). - _Wolfdieter Lang_, Nov 13 2007
%F A103371 From _Peter Bala_, Jan 24 2008: (Start)
%F A103371 Matrix product of A007318 and A122899.
%F A103371 O.g.f. for row n: (1-x)^n*P(n,1,0,(1+x)/(1-x)) = 1/(2*x)*(1-x)^(n+1)*( Legendre_P(n+1,(1+x)/(1-x)) - Legendre_P(n,(1+x)/(1-x)) ), where P(n,a,b,x) denotes the Jacobi polynomial.
%F A103371 O.g.f. for column k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. (End)
%F A103371 Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,1). (Cf. A194595, A197653, A197654). - _Peter Luschny_, Oct 20 2011
%F A103371 T(n,k) = A003056(n+1,k+1)*C(n,k)^2/(k+1). - _Peter Luschny_, Oct 29 2011
%F A103371 T(n,k) = A007318(n, k)*A135278(n, k), n >= k >= 0. - _Wolfdieter Lang_, Jul 31 2017
%F A103371 From _Natalia L. Skirrow_, Apr 14 2025: (Start)
%F A103371 T(n,k) = A008459(n,k) + n*N(n,k+1).
%F A103371 E.g.f.: e^(x*(1+y))*(I_0(2*x*sqrt(y)) + I_1(2*x*sqrt(y))/sqrt(y)), where I_n is the modified Bessel function of the first kind. (The I_0 contributes A008459(n,k), the I_1 contributes n*N(n,k+1))
%F A103371 O.g.f. for row n: (n+1)*2F1(-n,-n;2;y) = (n+1)*2F1(2+n,2+n;2;y)*(1-y)^(2*(n+1)) (by Euler's hypergeometric transformation); (n+1)*2F1(2+n,2+n;2;y) is the o.g.f. for row n of (k+n+1)!^2/(k!*(k+1)!*n!*(n+1)!), which is column n+1 of A132812.
%F A103371 O.g.f. for column k: 2F1(1+k,2+k;1;x)*x^k = 2F1(-k,-1-k;1;x)*x^k/(1-x)^(2+2*k). 2F1(-k,-1-k;1;x) is the kth row of A132813, the reflection of the kth row of this triangle.
%F A103371 O.g.f. for diagonal d (beginning at a(d,0)): (d+1)*x^d*2F1(d+1,d+2;2;x*y). 2F1(d+1,d+2;2;x) = 2F1(1-d,-d;2;x)/(1-x)^(2*d+1), numerator being the o.g.f. of row d of the Narayana triangle.
%F A103371 These respectively yield:
%F A103371 T(n,k) = Sum_{i=0..n+k} C(2*(n+1),i)*(-1)^i*A132812(n+1+k-i,n+1),
%F A103371 T(d+k,k) = Sum_{i=0..k} C(d-i+1+2*k,d-i)*T(k,k-i),
%F A103371 T(d+k,k) = Sum_{i=0..d} C(k-i + 2*d,k-i)*N(d,i+1)*(d+1).
%F A103371 E.g.f. for column k: 1F1(2+k;1;x)*x^k/k!.
%F A103371 E.g.f. for diagonal d: (d+1)*x^d*1F1(d+2;2;x*y)/d!. (End)
%e A103371 The triangle T(n, k) begins:
%e A103371 n\k 0 1 2 3 4 5 6 7 8 9 ...
%e A103371 0: 1
%e A103371 1: 2 1
%e A103371 2: 3 6 1
%e A103371 3: 4 18 12 1
%e A103371 4: 5 40 60 20 1
%e A103371 5: 6 75 200 150 30 1
%e A103371 6: 7 126 525 700 315 42 1
%e A103371 7: 8 196 1176 2450 1960 588 56 1
%e A103371 8: 9 288 2352 7056 8820 4704 1008 72 1
%e A103371 9: 10 405 4320 17640 31752 26460 10080 1620 90 1
%e A103371 ... reformatted. - _Wolfdieter Lang_, Jul 31 2017
%e A103371 From _R. J. Mathar_, Mar 29 2013: (Start)
%e A103371 The matrix inverse starts
%e A103371 1;
%e A103371 -2, 1;
%e A103371 9, -6, 1;
%e A103371 -76, 54, -12, 1;
%e A103371 1055, -760, 180, -20, 1;
%e A103371 -21906, 15825, -3800, 450, -30, 1;
%e A103371 636447, -460026, 110775, -13300, 945, -42, 1; (End)
%e A103371 O.g.f. of 4th diagonal [4, 40,200, ...] is G(3, x) = 4*(1 + 3*x + x^2)/(1 - x)^7, from the n = 3 row [1, 3, 1] of A001263. See a comment above. - _Wolfdieter Lang_, Jul 31 2017
%p A103371 A103371 := (n,k) -> binomial(n,k)^2*(n+1)/(k+1);
%p A103371 seq(print(seq(A103371(n, k), k=0..n)), n=0..7); # _Peter Luschny_, Oct 19 2011
%t A103371 Flatten[Table[Binomial[n,n-k]Binomial[n+1,n-k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 26 2014 *)
%t A103371 CoefficientList[Series[Series[E^(x(1+y))(BesselI[0,2*x*Sqrt[y]]+BesselI[1,2*x*Sqrt[y]]/Sqrt[y]),{x,0,8}],{y,0,8}],{x,y}]*Range[0,8]! (* _Natalia L. Skirrow_, Apr 14 2025 *)
%o A103371 (Maxima) create_list(binomial(n,k)*binomial(n+1,k+1),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */
%o A103371 (Haskell)
%o A103371 a103371 n k = a103371_tabl !! n !! k
%o A103371 a103371_row n = a103371_tabl !! n
%o A103371 a103371_tabl = map reverse a132813_tabl
%o A103371 -- _Reinhard Zumkeller_, Apr 04 2014
%o A103371 (Magma) /* As triangle */ [[Binomial(n,n-k)*Binomial(n+1,n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 01 2017
%o A103371 (PARI) for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n+1,k+1), ", "))) \\ _G. C. Greubel_, Nov 09 2018
%Y A103371 Cf. A008459, A122899, A194595, A197653.
%Y A103371 Cf. A007318, A000894 (central terms), A132813 (mirrored).
%Y A103371 Cf. A000027, A002411, A004302, A108647, A134287, A135278, A001263.
%K A103371 easy,nonn,tabl
%O A103371 0,2
%A A103371 _Paul Barry_, Feb 03 2005