A008967 -id:A008967 - cp's OEIS frontend

A117855 Number of nonzero palindromes of length n (in base 3).

2, 2, 6, 6, 18, 18, 54, 54, 162, 162, 486, 486, 1458, 1458, 4374, 4374, 13122, 13122, 39366, 39366, 118098, 118098, 354294, 354294, 1062882, 1062882, 3188646, 3188646, 9565938, 9565938, 28697814, 28697814, 86093442, 86093442, 258280326, 258280326, 774840978

Offset: 1

Martin Renner, May 02 2006

See A225367 for the sequence that counts all base 3 palindromes, including 0 (and thus also the number of n-digit terms in A006072). -- A nonzero palindrome of length L=2k-1 or of length L=2k is determined by the first k digits, which then determine the last k digits by symmetry. Since the first digit cannot be 0, there are 2*3^(k-1) possibilities. - M. F. Hasler, May 05 2013
From Gus Wiseman, Oct 18 2023: (Start)
Also the number of subsets of {1..n} with n not the sum of two subset elements (possibly the same). For example, the a(0) = 1 through a(4) = 6 subsets are:
  {}  {}   {}   {}     {}
      {1}  {2}  {1}    {1}
                {2}    {3}
                {3}    {4}
                {1,3}  {1,4}
                {2,3}  {3,4}
For subsets with no subset summing to n we have A365377.
Requiring pairs to be distinct gives A068911, complement A365544.
The complement is counted by A366131.
(End) [Edited by Peter Munn, Nov 22 2023]

			The a(3)=6 palindromes of length 3 are: 101, 111, 121, 202, 212, and 222. - _M. F. Hasler_, May 05 2013

Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A050683 and A070252.
Bisections are both A025192.
A093971/A088809/A364534 count certain types of sum-full subsets.
A108411 lists powers of 3 repeated, complement A167936.
Cf. A004526, A004737, A008967, A038754, A046663, A167762, A365376, A366130.

With[{c=NestList[3#&,2,20]},Riffle[c,c]] (* Harvey P. Dale, Mar 25 2018 *)
Table[Length[Select[Subsets[Range[n]],!MemberQ[Total/@Tuples[#,2],n]&]],{n,0,10}] (* Gus Wiseman, Oct 18 2023 *)

A117855(n)=2*3^((n-1)\2) \\ - M. F. Hasler, May 05 2013

def A117855(n): return 3**(n-1>>1)<<1 # Chai Wah Wu, Oct 28 2024

a(n) = 2*3^floor((n-1)/2).
a(n) = 2*A108411(n-1).
From Colin Barker, Feb 15 2013: (Start)
a(n) = 3*a(n-2).
G.f.: -2*x*(x+1)/(3*x^2-1). (End)

More terms from Colin Barker, Feb 15 2013

A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 5, 1, 3, 3, 1, 3, 6, 1, 3, 3, 6, 1, 3, 1, 3, 3, 3, 3, 6, 1, 3, 1, 3, 1, 6, 3, 3, 3, 3, 1, 5, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 5

Offset: 1

Gus Wiseman, Nov 06 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Is the image missing only 2 and 4?

			The prime indices of 15 are {2,3}, with sums of pairs:
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(15) = 3.
The prime indices of 180 are {1,1,2,2,3}, with sums of pairs:
  1+1 = 2
  1+2 = 3
  1+3 = 4
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(180) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Depends only on squarefree kernel A007947. (Even more exactly, on A322591 - Antti Karttunen, Jan 20 2025)
Positions of first appearances appear to be a subset of A325986.
For 2-element submultisets we have A366739, for all submultisets A299701.
A001222 counts prime factors (also indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A304793 counts positive subset-sums of prime indices.
A367096 lists semiprime divisors, count A086971.
Cf. A001248, A004526, A008967, A366737, A366738, A366740, A367093, A367097.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Total/@Tuples[prix[n],2]]],{n,100}]

A367095(n) = if(1==n, 0, my(pis=apply(primepi,factor(n)[,1]), pairsums = vector(binomial(1+#pis,2)), k=0); for(i=1,#pis,for(j=i,#pis,k++; pairsums[k] = pis[i]+pis[j])); #Set(pairsums)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

Original entry on oeis.org

1, 90, 630, 2310, 6930, 34650, 30030, 90090, 450450, 570570, 510510, 1531530, 7657650, 14804790, 11741730, 9699690, 29099070, 145495350

Offset: 0

Gus Wiseman, Nov 05 2023

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
Are all primorials after 210 included?

			The terms together with their prime indices begin:
       1: {}
      90: {1,2,2,3}
     630: {1,2,2,3,4}
    2310: {1,2,3,4,5}
    6930: {1,2,2,3,4,5}
   34650: {1,2,2,3,3,4,5}
   30030: {1,2,3,4,5,6}
   90090: {1,2,2,3,4,5,6}
  450450: {1,2,2,3,3,4,5,6}
  570570: {1,2,3,4,5,6,8}
  510510: {1,2,3,4,5,6,7}

The first part (semiprime divisors) is A086971, firsts A220264.
The second part (semi-sums of prime indices) is A366739, firsts A367097.
All sums of pairs of prime indices are counted by A367095.
The non-binary version is A367105.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts subset-sums of prime indices, positive A304793.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

nn=10000;
w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367093(n):
    for k in count(1):
        c, a = 0, set()
        for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
            if s not in a:
                a.add(s)
            else:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 13 2023

a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

a(12)-a(16) from Chai Wah Wu, Nov 13 2023

a(17) from Chai Wah Wu, Nov 18 2023

A240855 Number of partitions p of n into distinct parts including the number of parts.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 3, 5, 6, 8, 9, 10, 12, 16, 18, 22, 25, 29, 34, 41, 48, 55, 64, 74, 84, 98, 114, 130, 150, 170, 195, 222, 252, 287, 328, 371, 420, 475, 536, 604, 682, 766, 862, 970, 1088, 1218, 1365, 1526, 1704, 1904, 2124, 2366, 2637, 2934

Offset: 0

Clark Kimberling, Apr 14 2014

nonn

			a(10) counts these 4 partitions:  82, 631, 532, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from John Tyler Rascoe)
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.

Cf. A000009, A008967, A240861.

h:= (p, i)-> add(coeff(p, x, j)*x^j, j=i+1..degree(p)):
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 [h(g[1], i), g[2]])(b(n, i-1, p)+
     (f-> f+[0, coeff(f[1], x, i)])(b(n-i, min(n-i, i-1), p+1)))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..58);  # Alois P. Heinz, Mar 14 2024

z = 40;
f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)

p_q(k) = {prod(j=1,k, 1-q^j);}
mGB_q(N,M) = {p_q(N+M)/(p_q(M)*(p_q(N)^2));}
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=1, i, q^((i*(i+1)/2)+(j*(j-1))) * mGB_q(j-1,i-j))));
concat([0], Vec(g)) }
A_q(50) \\ John Tyler Rascoe, Mar 13 2024

a(n) = A000009(n) - A240861(n).

G.f.: Sum_{i>0} Sum_{j=1..i} q^((i*(i+1)/2) + j*(j-1)) * [j-1,i-j]q, where [N,M]_q = Product{j=1..N+M}(1-q^j) / (Product_{j=1..M}(1-q^j) * (Product_{j=1..N}(1-q^j))^2). - John Tyler Rascoe, Mar 13 2024

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Total/@Union[Subsets[#,{2}]]&]],{n,0,30}]

A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

Original entry on oeis.org

4, 6, 4, 9, 10, 4, 6, 14, 15, 4, 6, 9, 4, 10, 21, 22, 4, 6, 25, 26, 9, 4, 14, 6, 10, 15, 4, 33, 34, 35, 4, 6, 9, 38, 39, 4, 10, 6, 14, 21, 4, 22, 9, 15, 46, 4, 6, 49, 10, 25, 51, 4, 26, 6, 9, 55, 4, 14, 57, 58, 4, 6, 10, 15, 62, 9, 21, 4, 65, 6, 22, 33, 4, 34

Offset: 1

Gus Wiseman, Nov 08 2023

On the first interpretation, the first three rows are empty. On the second, the first row is (4).

			The semiprime divisors of 30 are {6,10,15}, so row 30 is (6,10,15). Without empty rows, this is row 19.
Triangle begins (empty rows indicated by dots):
   1: .
   2: .
   3: .
   4: 4
   5: .
   6: 6
   7: .
   8: 4
   9: 9
  10: 10
  11: .
  12: 4,6
Without empty rows:
   1: 4
   2: 6
   3: 4
   4: 9
   5: 10
   6: 4,6
   7: 14
   8: 15
   9: 4
  10: 6,9
  11: 4,10
  12: 21

For all divisors we have A027750.
Square terms are counted by A056170.
Row sums are A076290.
Squarefree terms are counted by A079275.
Row lengths are A086971, firsts A220264.
A000005 counts divisors.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A008967, A365829, A366738, A366739, A366740, A367093, A367098.

Table[Select[Divisors[n],PrimeOmega[#]==2&],{n,100}]

row(n) = select(x -> bigomega(x) == 2, divisors(n)); \\ Amiram Eldar, May 02 2025

A365659 Number of strict integer partitions of n that either have (1) length 2, or (2) greatest part n/2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 3, 4, 4, 6, 5, 8, 6, 10, 7, 12, 8, 15, 9, 18, 10, 21, 11, 25, 12, 29, 13, 34, 14, 40, 15, 46, 16, 53, 17, 62, 18, 71, 19, 82, 20, 95, 21, 109, 22, 125, 23, 144, 24, 165, 25, 189, 26, 217, 27, 248, 28, 283, 29, 324

Offset: 0

Gus Wiseman, Sep 16 2023

nonn

Also the number of strict integer partitions of n containing two possibly equal elements summing to n.

			The a(3) = 1 through a(11) = 5 partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (4,3)  (5,3)    (5,4)  (6,4)    (6,5)
                (4,1)  (5,1)    (5,2)  (6,2)    (6,3)  (7,3)    (7,4)
                       (3,2,1)  (6,1)  (7,1)    (7,2)  (8,2)    (8,3)
                                       (4,3,1)  (8,1)  (9,1)    (9,2)
                                                       (5,3,2)  (10,1)
                                                       (5,4,1)

Without repeated parts we have A140106.
The non-strict version is A238628.
For subsets instead of strict partitions we have A365544.
A000009 counts subsets summing to n.
A365046 counts combination-full subsets, differences of A364914.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A008967, A046663, A068911, A095944, A364272, A365376, A365377.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(Length[#]==2||Max@@#==n/2)&]], {n,0,30}]

from sympy.utilities.iterables import partitions
def A365659(n): return n>>1 if n&1 or n==0 else (m:=n>>1)+sum(1 for p in partitions(m) if max(p.values(),default=1)==1)-2 # Chai Wah Wu, Sep 18 2023

a(n) = (n-1)/2 if n is odd. a(n) = n/2 + A000009(n/2) - 2 if n is even and n > 0. - Chai Wah Wu, Sep 18 2023

A365828 Number of strict integer partitions of 2n not containing n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 148, 201, 270, 359, 475, 623, 811, 1050, 1351, 1728, 2201, 2789, 3517, 4418, 5527, 6887, 8553, 10585, 13055, 16055, 19685, 24065, 29343, 35685, 43287, 52387, 63253, 76200, 91605, 109897, 131575, 157231, 187539

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

			The a(0) = 1 through a(6) = 12 strict partitions:
  ()  (2)  (4)    (6)    (8)      (10)       (12)
           (3,1)  (4,2)  (5,3)    (6,4)      (7,5)
                  (5,1)  (6,2)    (7,3)      (8,4)
                         (7,1)    (8,2)      (9,3)
                         (5,2,1)  (9,1)      (10,2)
                                  (6,3,1)    (11,1)
                                  (7,2,1)    (5,4,3)
                                  (4,3,2,1)  (7,3,2)
                                             (7,4,1)
                                             (8,3,1)
                                             (9,2,1)
                                             (5,4,2,1)

The complement is counted by A111133.
For non-strict partitions we have A182616, complement A000041.
A000009 counts strict integer partitions.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A365827 counts strict partitions not of length 2, complement A140106.
Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

Table[Length[Select[IntegerPartitions[2n],UnsameQ@@#&&FreeQ[#,n]&]],{n,0,30}]

a(n) = A000009(2n) - A000009(n) + 1.

A115514 Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 4, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Roger L. Bagula, Mar 07 2006

T(n,k) = number of 2-element subsets of {1,2,...,n+2} such that the absolute difference of the elements is k+1, where 1 <= k < = n. E.g., T(7,3) = 3, the subsets are {1,5}, {2,6}, and {3,7}. - Christian Barrientos, Jun 27 2022

			Triangle begins as, for n >= 1, 1 <= k <= n,
  1;
  1, 1;
  2, 1, 1;
  2, 2, 1, 1;
  3, 2, 2, 1, 1;
  3, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000073, A004526, A008805, A008967.

Cf. A002620 (row sums), A008805 (diagonal sums), A142150 (alternating row sums)

[Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 14 2024

# Assuming offset 0:
Even := n -> (1 + (-1)^n)/2: # Iverson's even.
p := n -> add(add(Even(k)*x^j, j = 0..n-k), k = 0..n):
for n from 0 to 9 do seq(coeff(p(n), x, k), k=0..n) od; # Peter Luschny, Jun 03 2021

Table[Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[(n-k+2)//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

Sum_{k=1..n} T(n, k) = A002620(n+1) (row sums). - Gary W. Adamson, Oct 25 2007
T(n, k) = [x^k] p(n), where p(n) are partial Gaussian polynomials (A008967) defined by p(n) = Sum_{k=0..n} Sum_{j=0..n-k} even(k)*x^j, and even(k) = 1 if k is even and otherwise 0. We assume offset 0. - Peter Luschny, Jun 03 2021
T(n, k) = floor((n+2-k)/2). - Christian Barrientos, Jun 27 2022
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = A128623(n, k)/n.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A142150(n+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A008805(n-1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A002265(n+3). (End)

Edited by N. J. A. Sloane, Mar 23 2008 and Dec 15 2017

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)

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