A168625 -id:A168625 - cp's OEIS frontend

A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 11, 7, 1, 1, 9, 19, 19, 9, 1, 1, 11, 29, 39, 29, 11, 1, 1, 13, 41, 69, 69, 41, 13, 1, 1, 15, 55, 111, 139, 111, 55, 15, 1, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1, 1, 21, 109, 329, 659, 923, 923, 659, 329, 109, 21, 1

Offset: 0

Reinhard Zumkeller, Jun 20 2005

Eigensequence of the triangle = A001861. - Gary W. Adamson, Apr 17 2009

			Triangle begins as:
  1;
  1   1;
  1   3   1;
  1   5   5   1;
  1   7  11   7   1;
  1   9  19  19   9   1;
  1  11  29  39  29  11   1;
  1  13  41  69  69  41  13   1;
  1  15  55 111 139 111  55  15   1;
  1  17  71 167 251 251 167  71  17   1;
  1  19  89 239 419 503 419 239  89  19   1;

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

Cf. A000035, A000295, A001861, A005408, A014473, A028387, A030662.
Cf. A134760, A281362, A327767.
Cf. A000325 (row sums).
Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8).

a109128 n k = a109128_tabl !! n !! k
a109128_row n = a109128_tabl !! n
a109128_tabl = iterate (\row -> zipWith (+)
   ([0] ++ row) (1 : (map (+ 1) $ tail row) ++ [0])) [1]
-- Reinhard Zumkeller, Apr 10 2012

[2*Binomial(n,k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020

A109128 := proc(n,k)
    2*binomial(n,k)-1 ;
end proc: # R. J. Mathar, Jul 12 2016

Table[2*Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[2*binomial(n,k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 with T(n,0) = T(n,n) = 1.
Sum_{k=0..n} T(n, k) = A000325(n+1) (row sums).
T(n, k) = 2*binomial(n,k) - 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Sep 30 2007
T(n, 1) = 2*n - 1 = A005408(n+1) for n>0.
T(n, 2) = n^2 + n - 1 = A028387(n-2) for n>1.
T(n, k) = Sum_{j=0..n-k} C(n-k,j)*C(k,j)*(2 - 0^j) for k <= n. - Paul Barry, Apr 27 2006
T(n,k) = A014473(n,k) + A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
From G. C. Greubel, Apr 06 2024: (Start)
T(n, n-k) = T(n, k).
T(2*n, n) = A134760(n).
T(2*n-1, n) = A030662(n), for n >= 1.
Sum_{k=0..n-1} T(n, k) = A000295(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*[n=0] - A000035(n+1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A327767(n), for n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A281362(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A281362(n-1) - (1+(-1)^n)/2 for n >= 1.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) is the repeating pattern {1,1,0,-2,-3,-1,2,2,-1,-3,-2,0} with b(n) = b(n-12). (End)

Offset corrected by Reinhard Zumkeller, Apr 10 2012

A176203 Triangle read by rows: T(n, k) = 16*binomial(n, k) - 15.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 33, 33, 1, 1, 49, 81, 49, 1, 1, 65, 145, 145, 65, 1, 1, 81, 225, 305, 225, 81, 1, 1, 97, 321, 545, 545, 321, 97, 1, 1, 113, 433, 881, 1105, 881, 433, 113, 1, 1, 129, 561, 1329, 2001, 2001, 1329, 561, 129, 1, 1, 145, 705, 1905, 3345, 4017, 3345, 1905, 705, 145, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A007318 (q=0), A109128 (q=1), A131061 (q=2), A168625 (q=3), this sequence (q=4).
Row sums are: {1, 2, 19, 68, 181, 422, 919, 1928, 3961, 8042, 16219, ...}.
Former title: A recursive symmetrical triangular sequence:q=4: t(n, m, q) = 2*t(n, m, q-1) - 1. - G. C. Greubel, Mar 12 2020

			Triangle begins as:
  1;
  1,   1;
  1,  17,   1;
  1,  33,  33,    1;
  1,  49,  81,   49,    1;
  1,  65, 145,  145,   65,    1;
  1,  81, 225,  305,  225,   81,    1;
  1,  97, 321,  545,  545,  321,   97,    1;
  1, 113, 433,  881, 1105,  881,  433,  113,   1;
  1, 129, 561, 1329, 2001, 2001, 1329,  561, 129,   1;
  1, 145, 705, 1905, 3345, 4017, 3345, 1905, 705, 145, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8), this sequence (m=16).

[16*Binomial(n, k) -15: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020

A176203:= (n,k) -> 16*binomial(n, k) -15; seq(seq(A176203(n, k), k = 0..n), n = 0.. 12); # G. C. Greubel, Mar 12 2020

T[n_, m_, q]:= 2^q*(Binomial[n, m] -1) + 1; Table[T[n,m,4], {n,0,12}, {m,0,n} ]//Flatten (* modified by G. C. Greubel, Mar 12 2020 *)
Table[16*Binomial[n, k] -15, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[16*binomial(n, k) -15 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = binomial(n, m) and q = 4.
From G. C. Greubel, Mar 12 2020: (Start)
T(n, k, q) = 2^q * binomial(n, k) - (2^q - 1), with q = 4.
Sum_{k=0..n} T(n, k, q) = 2^(n + q) - (n + 1)*(2^q - 1) (row sums). (End)

Edited by G. C. Greubel, Mar 12 2020

A176204 Triangle T(n, k) = 4 * A008292(n+1, k) - 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 41, 41, 1, 1, 101, 261, 101, 1, 1, 225, 1205, 1205, 225, 1, 1, 477, 4761, 9661, 4761, 477, 1, 1, 985, 17169, 62473, 62473, 17169, 985, 1, 1, 2005, 58429, 352933, 624757, 352933, 58429, 2005, 1, 1, 4049, 191357, 1820765, 5241413, 5241413, 1820765, 191357, 4049, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

 This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A008292(n+1, k) (q=0), A176200 (q=1), this sequence (q=2).
Row sums are: {1, 2, 15, 84, 465, 2862, 20139, 161256, 1451493, 14515170, 159667167, ...}.
Former title: A recursive symmetrical triangular sequence based on Eulerian numbers: q=2: T(n, m, q) = 2*T(n, m, q-1) - 1.

			Triangle begins as:
  1;
  1,    1;
  1,   13,      1;
  1,   41,     41,       1;
  1,  101,    261,     101,       1;
  1,  225,   1205,    1205,     225,       1;
  1,  477,   4761,    9661,    4761,     477,       1;
  1,  985,  17169,   62473,   62473,   17169,     985,      1;
  1, 2005,  58429,  352933,  624757,  352933,   58429,   2005,    1;
  1, 4049, 191357, 1820765, 5241413, 5241413, 1820765, 191357, 4049, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A007318, A109128, A131061, A168625, A176200.

Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
[[4*Eulerian(n+1,k) -3: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 12 2020

A008292:= (n,k) -> add((-1)^j*binomial(n+1,j)*(k-j+1)^n, j=0..k+1);
A176204:= (n,k,q) -> 2^q*( A008292(n+1,k) -1) + 1;
seq(seq( A176204(n,k,2), k=0..n), n=0..12); # G. C. Greubel, Mar 12 2020

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
T[n_, m_, q_]:= 2^q*Eulerian[n+1, m] - 2^q +1;
Table[T[n, m, 2], {n,0,12}, {m,0,n}]//Flatten (* modified by G. C. Greubel, Mar 12 2020 *)

Eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n);
T(n,k,q) = 2^q*Eulerian(n+1,k) - (2^q - 1); \\ G. C. Greubel, Mar 12 2020

def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def T(n,k,q): return 2^q*Eulerian(n+1,k) - 2^q + 1
[[T(n,k,2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = A008292(n+1, m).
From G. C. Greubel, Mar 12 2020: (Start)
T(n, k, q) = 2^q * A008292(n+1, k) - (2^q - 1).
Sum_{k=0..n} T(n, k, q) = (n+1)*( 2^q * n! - 2^q + 1) (row sums). (End)

Edited by G. C. Greubel, Mar 12 2020

cp's OEIS Frontend

A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

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A176203 Triangle read by rows: T(n, k) = 16*binomial(n, k) - 15.

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A176204 Triangle T(n, k) = 4 * A008292(n+1, k) - 3, read by rows.

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