A071724 -id:A071724 - cp's OEIS frontend

A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).

1, 1, 2, 1, 3, 5, 1, 4, 9, 14, 1, 5, 14, 28, 42, 1, 6, 20, 48, 90, 132, 1, 7, 27, 75, 165, 297, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 1, 11, 65, 273, 910, 2548, 6188, 13260, 25194, 41990, 58786

Offset: 1

Wouter Meeussen

This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - Wolfdieter Lang, Jan 13 2006

			Triangle begins as:
  1;
  1, 2;
  1, 3,  5;
  1, 4,  9,  14;
  1, 5, 14,  28,  42;
  1, 6, 20,  48,  90,  132;
  1, 7, 27,  75, 165,  297,  429;
  1, 8, 35, 110, 275,  572, 1001, 1430;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862;

Reinhard Zumkeller, Rows n=1..151 of triangle, flattened
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
Wolfdieter Lang, First 10 rows.
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 6.

Alternate versions of (essentially) the same Catalan triangle: A009766, A033184, A047072, A059365, A099039, A106566, A130020.
Diagonals give A000108, A000245, A000344, A000588, A001392, A002057, A003517, A003518, A003519.
Row sums give A071724.
Cf. A009766, A350584.

a030237 n k = a030237_tabl !! n !! k
a030237_row n = a030237_tabl !! n
a030237_tabl = map init $ tail a009766_tabl
-- Reinhard Zumkeller, Jul 12 2012

[(n-k+1)*Binomial(n+k, k)/(n+1): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Mar 17 2021

A030237 := proc(n,m)
    (n-m+1)*binomial(n+m,m)/(n+1) ;
end proc: # R. J. Mathar, May 31 2016
# Compare the analogue algorithm for the Bell numbers in A011971.
CatalanTriangle := proc(len) local P, T, n; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([op(P), P[-1]]);
T := [op(T), P] od; T end: CatalanTriangle(6):
ListTools:-Flatten(%); # Peter Luschny, Mar 26 2022
# Alternative:
ogf := n -> (1 - 2*x)/(1 - x)^(n + 2):
ser := n -> series(ogf(n), x, n):
row := n -> seq(coeff(ser(n), x, k), k = 0..n-1):
seq(row(n), n = 1..11); # Peter Luschny, Mar 27 2022

T[n_, k_]:= T[n, k] = Which[k==0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1]];
Table[T[n, k], {n,1,12}, {k,0,n-1}] // Flatten (* Jean-François Alcover, Nov 14 2017 *)

T(n,k) = (n-k+1)*binomial(n+k, k)/(n+1) \\ Andrew Howroyd, Feb 23 2018

flatten([[(n-k+1)*binomial(n+k, k)/(n+1) for k in (0..n-1)] for n in (1..12)]) # G. C. Greubel, Mar 17 2021

T(n, k) = (n-k+1)*binomial(n+k, k)/(n+1).
Sum_{k=0..n-1} T(n,k) = A000245(n). - G. C. Greubel, Mar 17 2021
T(n, k) = [x^k] ((1 - 2*x)/(1 - x)^(n + 2)). - Peter Luschny, Mar 27 2022

Missing a(8) = T(7,0) = 1 inserted by Reinhard Zumkeller, Jul 12 2012

A002421 Expansion of (1-4*x)^(3/2) in powers of x.

Original entry on oeis.org

1, -6, 6, 4, 6, 12, 28, 72, 198, 572, 1716, 5304, 16796, 54264, 178296, 594320, 2005830, 6843420, 23571780, 81880920, 286583220, 1009864680, 3580429320, 12765008880, 45741281820, 164668614552, 595340375688, 2160865067312, 7871722745208, 28772503827312

Offset: 0

N. J. A. Sloane

Terms that are not divisible by 12 have indices in A019469. - Ralf Stephan, Aug 26 2004
From Ralf Steiner, Apr 06 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/8^k = 2F1(-3/2,-3/2,1,2).
Sum_{k>=0} a(k) / 2^k = -i. (End)

			G.f. = 1 - 6*x + 6*x^2 + 4*x^3 + 6*x^4 + 12*x^5 + 28*x^6 + 72*x^7 + 198*x^8 + 572*x^9 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424

Cf. A007054, A004001, A002420, A002422, A002423, A002424.

Cf. A000257, A001622, A071721, A071724, A085687.

Concatenation([1], List([1..40], n-> 12*Factorial(2*n-4) /( Factorial(n)*Factorial(n-2)) )) # G. C. Greubel, Jul 03 2019

[1,-6] cat [12*Catalan(n-2)/n: n in [2..30]]; // Vincenzo Librandi, Jun 11 2012

A002421 := n -> 3*4^(n-1)*GAMMA(-3/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002421(n), n=0..29); # Peter Luschny, Dec 14 2015

CoefficientList[Series[(1-4x)^(3/2),{x,0,40}],x] (* Vincenzo Librandi, Jun 11 2012 *)
a[n_]:= Binomial[ 3/2, n] (-4)^n; (* Michael Somos, Dec 04 2013 *)
a[n_]:= SeriesCoefficient[(1-4x)^(3/2), {x, 0, n}]; (* Michael Somos, Dec 04 2013 *)

{a(n) = binomial( 3/2, n) * (-4)^n}; /* Michael Somos, Dec 04 2013 */

{a(n) = if( n<0, 0, polcoeff( (1 - 4*x + x * O(x^n))^(3/2), n))}; /* Michael Somos, Dec 04 2013 */

((1-4*x)^(3/2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

a(n) = Sum_{m=0..n} binomial(n, m)*K_m(4), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) ~ (3/4)*Pi^(-1/2)*n^(-5/2)*2^(2*n)*(1 + 15/8*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
From Ralf Stephan, Mar 11 2004: (Start)
a(n) = 12*(2*n-4)! /(n!*(n-2)!), n > 1.
a(n) = 12*Cat(n-2)/n = 2(Cat(n-1) - 4*Cat(n-2)), in terms of Catalan numbers (A000108).
Terms that are not divisible by 12 have indices in A019469. (End)
Let rho(x)=(1/Pi)*(x*(4-x))^(3/2), then for n >= 4, a(n) = Integral_{x=0..4} (x^(n-4) *rho(x)) dx. - Groux Roland, Mar 16 2011
G.f.: (1-4*x)^(3/2) = 1 - 6*x + 12*x^2/(G(0) + 2*x); G(k) = (4*x+1)*k-2*x+2-2*x*(k+2)*(2*k+1)/G(k+1); for -1/4 <= x < 1/4, otherwise G(0) = 2*x; (continued fraction). - Sergei N. Gladkovskii, Dec 05 2011
G.f.: 1/G(0) where G(k) = 1 + 4*x*(2*k+1)/(1 - 1/(1 + (2*k+2)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012
G.f.: G(0)/2, where G(k) = 2 + 2*x*(2*k-3)*G(k+1)/(k+1). - Sergei N. Gladkovskii, Jun 06 2013 [Edited by Michael Somos, Dec 04 2013]
0 = a(n+2) * (a(n+1) - 14*a(n)) + a(n+1) * (6*a(n+1) + 16*a(n)) for all n in Z. - Michael Somos, Dec 04 2013
A232546(n) = 3^n * a(n). - Michael Somos, Dec 04 2013
G.f.: hypergeometric1F0(-3/2;;4*x). - R. J. Mathar, Aug 09 2015
a(n) = 3*4^(n-1)*Gamma(-3/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
From Ralf Steiner, Apr 06 2017: (Start)
Sum_{k>=0} a(k)/4^k = 0.
Sum_{k>=0} a(k)^2/16^k = 32/(3*Pi).
Sum_{k>=0} a(k)^2*(k/8)/16^k = 1/Pi.
Sum_{k>=0} a(k)^2*(-k/24+1/8)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(k-1/4)/16^k = 1/Pi.
Sum_{k>=0} a(k-1)^2*(2k-2)/16^k = 1/Pi.(End)
D-finite with recurrence: n*a(n) +2*(-2*n+5)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/3 + 10*Pi/(81*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 92/75 - 4*sqrt(5)*log(phi)/125, where phi is the golden ratio (A001622). (End)

A325189 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 1, 6, 2, 2, 0, 0, 0, 0, 7, 4, 2, 2, 0, 0, 0, 0, 6, 8, 4, 2, 2, 0, 0, 0, 0, 4, 12, 6, 4, 2, 2, 0, 0, 0, 0, 1, 15, 12, 6, 4, 2, 2, 0, 0, 0, 0, 0, 17, 15, 10, 6, 4, 2, 2

Offset: 0

Gus Wiseman, Apr 11 2019

The maximum origin-to-boundary graph-distance of an integer partition is one plus the maximum number of unit steps East or South in the Young diagram that can be followed, starting from the upper-left square, to reach a boundary square in the lower-right quadrant. It is also the side-length of the minimum triangular partition containing the diagram.

			Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  0  3  2
  0  0  0  3  2  2
  0  0  0  1  6  2  2
  0  0  0  0  7  4  2  2
  0  0  0  0  6  8  4  2  2
  0  0  0  0  4 12  6  4  2  2
  0  0  0  0  1 15 12  6  4  2  2
  0  0  0  0  0 17 15 10  6  4  2  2
  0  0  0  0  0 14 23 16 10  6  4  2  2
  0  0  0  0  0 10 30 23 14 10  6  4  2  2
  0  0  0  0  0  5 39 29 24 14 10  6  4  2  2
  0  0  0  0  0  1 42 42 31 22 14 10  6  4  2  2
Row 9 counts the following partitions:
  (432)   (54)     (63)      (72)       (81)        (9)
  (3321)  (333)    (621)     (711)      (21111111)  (111111111)
  (4221)  (441)    (6111)    (2211111)
  (4311)  (522)    (222111)  (3111111)
          (531)    (321111)
          (3222)   (411111)
          (5211)
          (22221)
          (32211)
          (33111)
          (42111)
          (51111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), table 1.
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Hook length and symplectic content in partitions, arXiv:2205.07322 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Graph Distance

Row sums are A000041. Column sums are A071724.

Cf. A065770, A096771, A115720, A115994, A139582, A325169, A325183, A325188, A325195, A325200, A366157.

otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(w=0); for(i=1, #p, w=max(w,#p-i+p[i])); r[w+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A366157(n). - Andrew Howroyd, Jan 12 2024

A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0, 0, 3, 2, 0, 2, 0, 1, 0, 6, 2, 0, 2, 0, 0, 4, 3, 4, 2, 0, 2, 0, 0, 6, 2, 6, 4, 2, 0, 2, 0, 0, 4, 9, 5, 4, 4, 2, 0, 2, 0, 1, 0, 15, 6, 8, 4, 4, 2, 0, 2, 0, 0, 5, 12, 12, 9, 6, 4, 4, 2, 0, 2, 0, 0, 10, 6, 21, 10, 12, 6, 4, 4, 2, 0, 2, 0

Offset: 0

Gus Wiseman, Apr 11 2019

			Triangle begins:
  1
  1  0
  0  2  0
  1  0  2  0
  0  3  0  2  0
  0  3  2  0  2  0
  1  0  6  2  0  2  0
  0  4  3  4  2  0  2  0
  0  6  2  6  4  2  0  2  0
  0  4  9  5  4  4  2  0  2  0
  1  0 15  6  8  4  4  2  0  2  0
  0  5 12 12  9  6  4  4  2  0  2  0
  0 10  6 21 10 12  6  4  4  2  0  2  0
  0 10 12 20 18 13 10  6  4  4  2  0  2  0
  0  5 27 20 23 16 16 10  6  4  4  2  0  2  0
  1  0 38 22 32 22 19 14 10  6  4  4  2  0  2  0
  0  6 34 38 34 35 20 22 14 10  6  4  4  2  0  2  0
  0 15 22 57 44 40 34 23 20 14 10  6  4  4  2  0  2  0
  0 20 20 71 55 54 45 32 26 20 14 10  6  4  4  2  0  2  0
  0 15 43 70 71 66 60 44 35 24 20 14 10  6  4  4  2  0  2  0
  0  6 74 64 99 83 70 65 42 38 24 20 14 10  6  4  4  2  0  2  0
Row n = 9 counts the following partitions (empty columns not shown):
  (432)   (333)    (54)      (63)      (72)       (81)        (9)
  (3321)  (441)    (621)     (6111)    (711)      (21111111)  (111111111)
  (4221)  (522)    (22221)   (222111)  (2211111)
  (4311)  (531)    (51111)   (411111)  (3111111)
          (3222)   (321111)
          (5211)
          (32211)
          (33111)
          (42111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Row sums are A000041. Column k = 1 is A325191. Column k = 2 is A325199.
T(n,k) = A325189(n,k) - A325188(n,k).
Cf. A046660, A065770, A071724, A243055, A325169, A325178, A325195, A325196, A325197, A366157, A368986.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otbmax[#]-otb[#]==k&]],{n,0,20},{k,0,n}]

row(n)={my(r=vector(n+1)); forpart(p=n, my(b=#p,c=0); for(i=1, #p, my(x=#p-i+p[i]); b=min(b,x); c=max(c,x)); r[c-b+1]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A366157(n) - A368986(n). - Andrew Howroyd, Jan 13 2024

A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 1, 2, 0, 3, 2, 1, 1, 4, 1, 5, 2, 1, 3, 6, 1, 7, 1, 2, 3, 8, 2, 2, 4, 2, 2, 9, 0, 10, 4, 3, 5, 2, 2, 11, 6, 4, 2, 12, 1, 13, 3, 1, 7, 14, 3, 3, 1, 5, 4, 15, 2, 3, 2, 6, 8, 16, 1, 17, 9, 1, 5, 4, 2, 18, 5, 7, 1, 19, 3, 20, 10, 1, 6, 3, 3, 21, 3, 3, 11

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition (3,3) has Heinz number 25 and diagram
  o o o
  o o o
containing maximal triangular partition
  o o
  o
and contained in minimal triangular partition
  o o o o
  o o o
  o o
  o
so a(25) = 4 - 2 = 2.

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, A325199, A325200.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]],{n,100}]

A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

Original entry on oeis.org

0, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 10, 10, 5, 0, 6, 15, 20, 15, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 28, 56, 70, 56, 28, 8, 0, 9, 36, 84, 126, 126, 84, 36, 9, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 11, 55, 165, 330, 462

Offset: 0

Gus Wiseman, Apr 11 2019

The Heinz numbers of these partitions are given by A325196.

Under the Bulgarian solitaire step, these partitions form cycles of length >= 2. Length >= 2 means not the length=1 self-loop which occurs from the triangular partition when n is a triangular number. See A074909 for self-loops included. - Kevin Ryde, Sep 27 2019

			The a(2) = 2 through a(12) = 10 partitions (empty columns not shown):
  (2)   (22)   (32)   (322)   (332)   (432)   (4322)   (4332)
  (11)  (31)   (221)  (331)   (422)   (3321)  (4331)   (4422)
        (211)  (311)  (421)   (431)   (4221)  (4421)   (4431)
                      (3211)  (3221)  (4311)  (5321)   (5322)
                              (3311)          (43211)  (5331)
                              (4211)                   (5421)
                                                       (43221)
                                                       (43311)
                                                       (44211)
                                                       (53211)

FindStat, St000380: Half the perimeter of the largest rectangle that fits inside the diagram of an integer partition
FindStat, St000384: The maximal part of the shifted composition of an integer partition
FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
Eric Weisstein's World of Mathematics, Graph Distance

Column k=1 of A325200.

Cf. A060687, A065770, A071724, A256617, A325166, A325169, A325178, A325179, A325181, A325187, A325188, A325189, A325195, A325196.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]+1==otbmax[#]&]],{n,0,30}]

a(n) = my(t=ceil(sqrtint(8*n+1)/2), r=n-t*(t-1)/2); if(r==0,0, binomial(t,r)); \\ Kevin Ryde, Sep 27 2019

Positions of zeros are A000217 = n * (n + 1) / 2.

a(n) = A074909(n) - A010054(n). - Kevin Ryde, Sep 27 2019

A325193 Number of integer partitions whose sum plus co-rank is n, where co-rank is maximum of length and largest part.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 2, 5, 5, 8, 8, 14, 14, 22, 24, 35, 39, 54, 62, 84, 97, 127, 148, 192, 224, 284, 334, 418, 492, 610, 716, 880, 1035, 1259, 1480, 1790, 2100, 2522, 2958, 3533, 4135, 4916, 5742, 6798, 7928, 9344, 10878, 12778, 14846, 17378, 20156, 23520

Offset: 0

Gus Wiseman, Apr 12 2019

nonn

			The a(4) = 2 through a(12) = 14 partitions:
  (2)   (21)  (3)    (31)   (4)     (33)    (5)      (43)     (6)
  (11)        (22)   (211)  (32)    (41)    (42)     (51)     (44)
              (111)         (221)   (222)   (322)    (332)    (52)
                            (311)   (321)   (331)    (421)    (333)
                            (1111)  (2111)  (411)    (2221)   (422)
                                            (2211)   (3211)   (431)
                                            (3111)   (4111)   (511)
                                            (11111)  (21111)  (2222)
                                                              (3221)
                                                              (3311)
                                                              (4211)
                                                              (22111)
                                                              (31111)
                                                              (111111)

FindStat, St000784: The maximum of the length and the largest part of the integer partition

Row sums of A325194.

Cf. A051924, A065770, A071724, A096771, A115720, A252464, A257990, A263297, A325178, A325189, A325192.

Table[Sum[Length[Select[IntegerPartitions[k],Max[Length[#],Max[#]]==n-k&]],{k,0,n}],{n,0,30}]

A366431 G.f. A(x) satisfies A(x) = 1 + x * (A(x) / (1 - x))^(5/2).

Original entry on oeis.org

1, 1, 5, 25, 135, 775, 4651, 28845, 183450, 1190050, 7844230, 52389678, 353770190, 2411324700, 16568343325, 114639216915, 798076174113, 5586035989185, 39287407321075, 277508001643575, 1967816928168265, 14003018984540741, 99965175670335750

Offset: 0

Seiichi Manyama, Oct 09 2023

nonn

Partial sums give A366400.

Cf. A006319, A071724, A213282, A213336, A366432, A366433, A366434, A366435, A366436, A366437.

a(n) = sum(k=0, n, binomial(n+3*k/2-1, n-k)*binomial(5*k/2, k)/(3*k/2+1));

a(n) = Sum_{k=0..n} binomial(n+3*k/2-1,n-k) * binomial(5*k/2,k) / (3*k/2+1).

A115142 Third convolution of A115140.

Original entry on oeis.org

1, -3, 0, -1, -3, -9, -28, -90, -297, -1001, -3432, -11934, -41990, -149226, -534888, -1931540, -7020405, -25662825, -94287120, -347993910, -1289624490, -4796857230, -17902146600, -67016296620, -251577050010, -946844533674, -3572042254128, -13505406670700

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1668

Cf. A000108, A000245, A071724.

Cf. A115139, A115140, A115141, A115143 - A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2*(1-2*x)-(1-x)*(1-Sqrt(1-4*x)))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(2*(1-2*x)-(1-x)*(1-Sqrt[1-4*x]))/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2) \\ G. C. Greubel, Feb 12 2019

((2*(1-2*x)-(1-x)*(1-sqrt(1-4*x)))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^3 = P(4, x) - x*P(3, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(4, x)=1-2*x and P(3, x)=1-x.
a(n) = -C3(n-3), n >= 3, with C3(n):= A000245(n+1) (third convolution of Catalan numbers). a(0)=1, a(1)=-3, a(2)=0. [1, -3] is the row n=3 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-3)*a(n) -2*(n-2)*(2*n-5)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
O.g.f.: (1/8)*(1 + sqrt(1 - 4*x))^3 = ( hypergeom([-1/4, -3/4], [-1/2], 4*x) )^2. - Peter Bala, Mar 04 2022

A158909 Riordan array (1/((1-x)(1-x^2)), x/(1-x)^2). Triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 7, 5, 1, 3, 13, 16, 7, 1, 3, 22, 40, 29, 9, 1, 4, 34, 86, 91, 46, 11, 1, 4, 50, 166, 239, 174, 67, 13, 1, 5, 70, 296, 553, 541, 297, 92, 15, 1, 5, 95, 496, 1163, 1461, 1068, 468, 121, 17, 1, 6, 125, 791, 2269, 3544, 3300, 1912, 695, 154, 19, 1

Offset: 0

Paul Barry, Mar 30 2009

Diagonal sums are the Jacobsthal numbers A001045.
Transforms r^n into the symmetric third-order sequence with g.f. 1/(1-(r+1)x-(r+1)x^2+x^3), see the formulas.
From Wolfdieter Lang, Oct 22 2019: (Start)
The signed triangle t(n, k) = (-1)^(n-k)*T(n, k) appears in the expansion [n+2, 2]q / q^n = Sum{k=0} t(n, k)*y^(2*k), with y = q^(1/2) + q^(-1/2), where [n+2, 2]_q are q-binomial coefficients (see A008967, but with a different offset). The formula is [n+2, 2]_q / q^n = S(n+1, y)*S(n, y)/y with Chebyshev S polynomials (A049310). This is a polynomial in y^2 but not in q after replacement of the given y = y(q).
The A-sequence for this Riordan triangle is A(n) = (-1)^n*A115141(n) with o.g.f A(x) = 1 + x*(1 + c(-x)), with c(x) generating A000108 (Catalan).
The Z-sequence is z(n) = (-1)^(n+1)*A071724(n), for n >= 1 and z(0) = 1. The o.g.f. is Z(x) = 1 + x*c(-x)^3. See A071724 for a link on A- and Z-sequences, and their use for the recurrence. (End)
T(n,k) is the number of tilings of a (2*n+1)-board (a 1 X (2*n+1) rectangular board) using 2*k+1 squares and 2*(n-k) (1,1)-fences. A (1,1)-fence is a tile composed of two squares separated by a gap of width 1. - Michael A. Allen, Mar 20 2021

			From _Wolfdieter Lang_, Oct 22 2019: (Start)
The triangle T(n, k) begins:
  n\k  0   1   2    3    4    5    6   7   8  9 10 ...
  ----------------------------------------------------
  0:   1
  1:   1   1
  2:   2   3   1
  3:   2   7   5    1
  4:   3  13  16    7    1
  5:   3  22  40   29    9    1
  6:   4  34  86   91   46   11    1
  7:   4  50 166  239  174   67   13   1
  8:   5  70 296  553  541  297   92  15   1
  9:   5  95 496 1163 1461 1068  468 121  17  1
  10:  6 125 791 2269 3544 3300 1912 695 154 19  1
  ...
----------------------------------------------------------------------------
Recurrence: T(5, 2) = 16 + 13 + 5 + 7 - 1 = 40, and T(5, 0) = 3 + 2 - 2 = 3. [using _Philippe Deléham_'s Nov 12 2013 recurrence]
Recurrence from A-sequence [1, 2, -1, 2, -5, ...]: T(5, 2) = 1*13 + 2*16 - 1*7 + 2*1 = 40.
Recurrence from Z-sequence [1, 1, -3, 9, -28, ...]: T(5, 0) = 1*3 + 1*13 - 3*16 + 9*7 - 28*1 = 3. (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, JIS 24 (2021) 21.3.8.

Cf. A000045, A001654 (row sums), A008967, A035317, A049310, A092879, A335964.

[(&+[(-1)^(j+n-k)*Binomial(2*k+j+1, j): j in [0..n-k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2021

T := (n,k) -> binomial(k+n+2, n-k+1)*hypergeom([1, k+n+3], [n-k+2], -1) + (-1)^(n-k)/4^(k+1):
seq(seq(simplify(T(n,k)), k=0..n), n=0..9); # Peter Luschny, Oct 31 2019

Table[Sum[(-1)^(j+n-k)*Binomial[j+2*k+1, j], {j,0,n-k}], {n,0,12}, {k,0,n}] // Flatten (* G. C. Greubel, Mar 18 2021 *)

flatten([[sum((-1)^(j+n-k)*binomial(j+2*k+1, j) for j in (0..n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2021

Sum_{k=0..n} T(n,k) = Fibonacci(n+1)*Fibonacci(n+2) = A001654(n+1).
From Johannes W. Meijer, Jul 20 2011: (Start)
T(n, k) = Sum_{i=0..n-k} (-1)^(i+n-k) * binomial(i+2*k+1, i).
T(n, k) = A035317(n+k, n-k) = A092879(n, n-k).
Sum_{k=0..n} T(n, k)*r^k = coeftayl(1/(1-(r+1)*x-(r+1)*x^2+x^3), x=0, n). [Barry] (End)
T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) - T(n-3, k), T(0, 0) = 1, T(n, k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013
From Wolfdieter Lang, Oct 22 2019: (Start)
O.g.f. for the row polynomials (that is for the triangle): G(z, x) = 1/((1 + z)*(1 - (x + 2)*z + z^2)), and
O.g.f. for column k: x^k/((1+x)*(1-x)^(2*(k+1))) (Riordan property). (End)
T(n, k) = binomial(k + n + 2, n - k + 1)*hypergeom([1, k + n + 3], [n - k + 2], -1) + (-1)^(n - k)/4^(k + 1). - Peter Luschny, Oct 31 2019
From Michael A. Allen, Mar 20 2021: (Start)
T(n,k) = A335964(2*n+1,n-k).
T(n,k) = T(n-2,k) + binomial(n+k,2*k). (End)

cp's OEIS Frontend

A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n).

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A002421 Expansion of (1-4*x)^(3/2) in powers of x.

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A325189 Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.

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A325200 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.

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A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

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A325191 Number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

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