A157898 -id:A157898 - cp's OEIS frontend

A157901 Triangle read by rows: A000012 * A157898.

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

A097076 Expansion of g.f. x/(1 - x - 3*x^2 - x^3).

Original entry on oeis.org

0, 1, 1, 4, 8, 21, 49, 120, 288, 697, 1681, 4060, 9800, 23661, 57121, 137904, 332928, 803761, 1940449, 4684660, 11309768, 27304197, 65918161, 159140520, 384199200, 927538921, 2239277041, 5406093004, 13051463048, 31509019101, 76069501249, 183648021600

Offset: 0

Paul Barry, Jul 22 2004

Counts walks of length n between two vertices of a triangle, when a loop has been added at the third vertex.
a(n) is the center term of the 3 X 3 matrix [0,1,0; 0,0,1; 1,3,1]^n. - Gary W. Adamson, May 30 2008
Starting (1, 1, 4, 8, 21, ...) = row sums of triangle A157898. - Gary W. Adamson, Mar 08 2009
Convolution of Pell(n) = A000129(n) and (-1)^n. - Paul Barry, Oct 22 2009
a(n+1) is the number of ways to choose points on a 2 X n lattice eliminating the upper left and lower right corners such that the points are not adjacent to each other. (See A375726 for proof) - Yifan Xie, Aug 25 2024
a(n+1) is the number of compositions (ordered partitions) of n into parts 1, 2, and 3 where there are three kinds of part 2. - Joerg Arndt, Aug 27 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Bodeen, S. Butler, T. Kim, X. Sun and S. Wang, Tiling a strip with triangles, El. J. Combinat. 21 (1) (2014) P1.7.
Mark Shattuck, Combinatorial Proofs of Some Formulas for Triangular Tilings, Journal of Integer Sequences, 17 (2014), #14.5.5.
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A051927, A097075, A110048, A157898, A375726.

[(Evaluate(DicksonFirst(n,-1), 2) -2*(-1)^n)/4: n in [0..40]]; // G. C. Greubel, Aug 18 2022

CoefficientList[Series[x/(1-x-3x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{1,3,1},{0,1,1},40]  (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

[(lucas_number2(n,2,-1) -2*(-1)^n)/4 for n in (0..40)] # G. C. Greubel, Aug 18 2022

a(n) = ( (1+sqrt(2))^n + (1-sqrt(2))^n - 2*(-1)^n )/4.
a(n) = a(n-1) + 3*a(n-2) + a(n-3). [corrected by Paul Curtz, Mar 04 2008]
a(n) = (Sum_{k=0..floor(n/2)} binomial(n, 2*k)*2^k)/2 - (-1)^n/2.
a(n) = (A001333(n) - (-1)^n)/2.
a(n) = Sum_{k=0..n} (-1)^k*Pell(n-k). - Paul Barry, Oct 22 2009
From R. J. Mathar, Jul 06 2011: (Start)
G.f.: x / ( (1+x)*(1-2*x-x^2) ).
a(n) + a(n+1) = A000129(n+1). (End)
E.g.f.: (exp(x)*cosh(sqrt(2)*x) - cosh(x) + sinh(x))/2. - Stefano Spezia, Mar 31 2024

cp's OEIS Frontend

A157901 Triangle read by rows: A000012 * A157898.

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