A079926 -id:A079926 - cp's OEIS frontend

A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 15, 21, 15, 5, 8, 30, 53, 53, 30, 8, 13, 58, 124, 157, 124, 58, 13, 21, 109, 273, 417, 417, 273, 109, 21, 34, 201, 577, 1029, 1239, 1029, 577, 201, 34, 55, 365, 1181, 2405, 3375, 3375, 2405, 1181, 365, 55, 89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89

Offset: 0

Christian G. Bower, Jan 19 2004

T(n,k) is the number of lattice paths from (0,0) to (k,n-k) using steps (1,0),(2,0),(0,1),(0,2),(1,1). - Seiichi Manyama, Apr 26 2025.

			This triangle begins:
   1;
   1,   1;
   2,   3,    2;
   3,   7,    7,    3;
   5,  15,   21,   15,    5;
   8,  30,   53,   53,   30,     8;
  13,  58,  124,  157,  124,    58,   13;
  21, 109,  273,  417,  417,   273,  109,   21;
  34, 201,  577, 1029, 1239,  1029,  577,  201,   34;
  55, 365, 1181, 2405, 3375,  3375, 2405, 1181,  365,  55;
  89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89;
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Row sums: A015518(n+1). Columns 0-1: A000045(n+1), A023610(n-1).
Cf. A090174, A212338 (column 2), A192364 (central terms).
Cf. A036355.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
     `if`(n<1, 1, add(add(T(n-i, k-j), j=0..i), i=1..2)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 14 2022

A091533[-2, n2_] = 0; A091533[n1_, -2] = 0; A091533[-1, n2_] = 0; A091533[n1_, -1] = 0; A091533[0, 0] = 1; A091533[n1_, n2_] := A091533[n1, n2] = A091533[n1 - 1, n2] + A091533[n1, n2 - 1] + A091533[n1 - 1, n2 - 1] + A091533[n1 - 2, n2] + A091533[n1, n2 - 2]; Table[A091533[x - y, y], {x, 0, 9}, {y, 0, x}] // Flatten (* Robert P. P. McKone, Jan 14 2022 *)

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.
G.f.: A(x, y) = 1/(1-x-x*y-x^2-x^2*y-x^2*y^2).
Sum_{k = 0..n} T(n,k)*x^k = A000045(n+1), A015518(n+1), A015524(n+1), A200069(n+1) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 30 2013
Sum_{k = 0..floor(n/2)} T(n-k,k) = (-1)^n*A079926(n). - Philippe Deléham, Oct 30 2013

A248944 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in i..i+k and no value appearing more than 1 time.

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 5, 13, 14, 5, 6, 21, 36, 26, 6, 7, 31, 76, 90, 46, 7, 8, 43, 140, 246, 212, 79, 8, 9, 57, 234, 566, 738, 478, 133, 9, 10, 73, 364, 1146, 2104, 2108, 1044, 221, 10, 11, 91, 536, 2106, 5150, 7364, 5794, 2227, 364, 11, 12, 111, 756, 3590, 11196, 21652, 24720

Offset: 1

R. H. Hardin, Oct 17 2014

Table starts
..2...3....4......5......6.......7........8........9........10........11
..3...7...13.....21.....31......43.......57.......73........91.......111
..4..14...36.....76....140.....234......364......536.......756......1030
..5..26...90....246....566....1146.....2106.....3590......5766......8826
..6..46..212....738...2104....5150....11196....22162.....40688.....70254
..7..79..478...2108...7364...21652....55532...127604....268108....523244
..8.133.1044...5794..24720...86608...260720...693552...1666000...3675680
..9.221.2227..15458..80196..334072..1173240..3598120...9856552..24553080
.10.364.4664..40296.253072.1249768..5112544.17990600..56010096.157175032
.11.596.9627.103129.780902.4557284.21670160.87396728.308055528.971055240

R. H. Hardin, Table of n, a(n) for n = 1..507

Column 1 is A000027(n+1)
Column 2 is A001924(n+1)
Column 3 is A079922
Column 4 is A079923
Column 5 is A079924
Column 6 is A079925
Column 7 is A079926
Row 1 is A000027(n+1)
Row 2 is A002061(n+1)
Row 3 is A061989(n+3)
Row 4 is A079909
Row 5 is A079910
Row 6 is A079911
Row 7 is A079912

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-2)
k=2: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
k=3: a(n) = 4*a(n-1) -4*a(n-2) -2*a(n-4) +4*a(n-5) -a(n-8)
k=4: [order 16]
k=5: [order 32]
k=6: [order 63]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + n + 1
n=3: a(n) = n^3 + 3*n
n=4: a(n) = n^4 - 2*n^3 + 9*n^2 - 8*n + 6 for n>1
n=5: a(n) = n^5 - 5*n^4 + 25*n^3 - 55*n^2 + 80*n - 46 for n>1
n=6: a(n) = n^6 - 9*n^5 + 60*n^4 - 225*n^3 + 555*n^2 - 774*n + 484 for n>3
n=7: a(n) = n^7 - 14*n^6 + 126*n^5 - 700*n^4 + 2625*n^3 - 6342*n^2 + 9072*n - 5840 for n>4

cp's OEIS Frontend

A091533 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,1.

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