A090749 -id:A090749 - cp's OEIS frontend

A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

0, 1, -1, 2, 0, -2, 3, 2, -2, -3, 4, 5, 0, -5, -4, 5, 9, 5, -5, -9, -5, 6, 14, 14, 0, -14, -14, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10

Offset: 0

Reinhard Zumkeller, Jul 12 2012

			The triangle begins:
    0:                              0
    1:                            1   -1
    2:                          2   0   -2
    3:                       3    2   -2   -3
    4:                     4    5   0   -5   -4
    5:                  5    9    5   -5   -9   -5
    6:                6   14   14   0  -14  -14   -6
    7:             7   20   28   14  -14  -28  -20   -7
    8:           8   27   48   42   0  -42  -48  -27   -8
    9:        9   35   75   90   42  -42  -90  -75  -35   -9
   10:     10   44  110  165  132   0 -132 -165 -110  -44  -10
   11:  11   54  154  275  297  132 -132 -297 -275 -154  -54  -11  .

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A000004, A000096, A000108, A000245, A000344, A000588, A000589, A000590, A001392, A002057, A003517, A003518, A003519, A005557, A005586, A005587, A008313, A014495, A064059, A064061, A080956, A090749, A097808, A112467, A124087, A124088, A129936, A259525.

a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
   where diff row = zipWith (-) (tail row) row

row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)

T(n,k) = A007318(n+1,k+1) - A007318(n+1,k), 0<=k<=n, i.e. first differences of rows in Pascal's triangle;
T(n,k) = -T(n,k);
row sums and central terms equal 0, cf. A000004;
sum of positive elements of n-th row = A014495(n+1);
T(n,0) = n;
T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;
T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);
T(n,3) = A005587(n-6) for n > 5;
T(n,4) = A005557(n-9) for n > 8;
T(n,5) = A064059(n-11) for n > 10;
T(n,6) = A064061(n-13) for n > 12;
T(n,7) = A124087(n) for n > 14;
T(n,8) = A124088(n) for n > 16;
T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;
T(2*n+3,n) = A000245(n+2);
T(2*n+4,n) = A002057(n+1);
T(2*n+5,n) = A000344(n+3);
T(2*n+6,n) = A003517(n+3);
T(2*n+7,n) = A000588(n+4);
T(2*n+8,n) = A003518(n+4);
T(2*n+9,n) = A001392(n+5);
T(2*n+10,n) = A003519(n+5);
T(2*n+11,n) = A000589(n+6);
T(2*n+12,n) = A090749(n+6);
T(2*n+13,n) = A000590(n+7).

A050144 T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 3, 4, 1, 5, 9, 14, 6, 1, 14, 28, 48, 27, 8, 1, 42, 90, 165, 110, 44, 10, 1, 132, 297, 572, 429, 208, 65, 12, 1, 429, 1001, 2002, 1638, 910, 350, 90, 14, 1, 1430, 3432, 7072, 6188, 3808, 1700, 544, 119, 16, 1

Offset: 0

Clark Kimberling

Let V=(e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h)=(e(1),...,e(h)) and m(h)=(#1's in V(h))-(#0's in V(h)) for h=1,...,n. Then M(p,q,r)=number of V having r=max{m(h)}.
The interpretation of T(n,k) as RU walks in terms of M(.,.,.) in the NAME is erroneous. There seems to be a pattern along subdiagonals:
 M(3,1,1) = 4  = T(3,2); M(3,1,2) = 1  = T(4,4); M(5,2,1) = 20 = T(5,3); M(5,2,2) = 7  = T(6,5); M(5,2,3) = 1  = T(7,7); M(7,3,0) = 165 = T(6,2); M(7,3,1) = 110 = T(7,4); M(7,3,2) = 44  = T(8,6); M(7,3,3) = 10  = T(9,8); M(7,3,4) = 1   = T(10,10); M(9,4,0) = 1001 = T(8,3); M(9,4,1) = 637  = T(9,5); M(9,4,2) = 273  = T(10,7); M(9,4,3) = 77   = T(11,9); M(9,4,4) = 13   = T(12,11); M(9,4,5) = 1   = T(13,13); M(11,5,0) = 6188 = T(10,4); M(11,5,1) = 3808 = T(11,6); M(11,5,2) = 1700 = T(12,8); M(11,5,3) = 544  = T(13,...); M(11,5,4) = 119; M(11,5,5) = 16; M(11,5,6) = 1; M(13,6,0) = 38760 = T(12,5); M(13,6,1) = 23256 = T(13,7); M(13,6,2) = 10659 = T(14,9); - R. J. Mathar, Jul 31 2024

			Triangle begins:
     0
     1    0
     1    1    1
     2    3    4    1
     5    9   14    6    1
    14   28   48   27    8    1
    42   90  165  110   44   10    1
   132  297  572  429  208   65   12    1
   429 1001 2002 1638  910  350   90   14    1
  1430 3432 7072 6188 3808 1700  544  119   16    1

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

{M(2n, 0, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.

Cf. A033184, A050153, A000108 (column 0), A000245 (column 1), A002057 (column 2), A000344 (column 3), A003517 (column 4), A000588 (column 5), A003518 (column 6), A001392 (column 7), A003519 (column 8), A000589 (column 9), A090749 (column 10).

A050144 := proc(n,k)
    if n < k then
        0;
    elif k =0 then
        if n =0 then
            0 ;
        else
            A000108(n-1) ;
        end if;
    elif k = 1 then
        add( procname(n-1-j,0)*A000108(j+1),j=0..n-1) ;
    elif k = 2 then
        add( procname(n-j,1)*A000108(j),j=0..n) ;
    else
        add( procname(n-1-j,k-1)*A000108(j),j=0..n-1) ;
    end if;
end proc:
seq(seq( A050144(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 30 2024

c[n_] := Binomial[2 n, n]/(n + 1);
t[n_, k_] := Which[k == 0, c[n - 1],
  k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
  k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
  k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
t[0, 0] = 0;
Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
(* Clark Kimberling, Jul 30 2024 *)

For n > 0: Sum_{k>=0} T(n, k) = binomial(2*n-1, n); see A001700. - Philippe Deléham, Feb 13 2004 [Erroneous sum-formula deleted. R. J. Mathar, Jul 31 2024]
T(n, k)=0 if n < k; T(0, 0)=0, T(n, 0) = A000108(n-1) for n > 0; T(n, 1) = Sum_{j>=0} T(n-1-j, 0)*A000108(j+1); T(n, 2) = Sum_{j>=0} T(n-j, 1)*A000108(j); for k > 2, T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*A000108(j+1). - Philippe Deléham, Feb 13 2004 [Corrected by Sean A. Irvine, Aug 08 2021]
For the column k=0, g.f.: x*C(x); for the column k=1, g.f.: x*C(x)*(C(x)-1); for the column k, k > 1, g.f.: x*C(x)^2*(C(x)-1)^(k-1); where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108. - Philippe Deléham, Feb 13 2004
T(n,0) = A033184(n,2). T(n,1) = A033184(n+1,3), T(n,k) = A033184(n+2,k+2) for k>=2. - R. J. Mathar, Jul 31 2024

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Philippe Deléham, Feb 01 2014

Row sums are A109262(n+1).

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022

cp's OEIS Frontend

A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.

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A050144 T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.

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A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

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