A001788 -id:A001788 - cp's OEIS frontend

A236999 Odd part of n*(n+3)/2-1 (A034856).

1, 1, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 169, 47, 13, 229, 251, 137, 149, 323, 349, 47, 101, 433, 463, 247, 263, 559, 593, 157, 83, 701, 739, 389, 409, 859, 901, 59, 247, 1033, 1079, 563, 587, 1223, 1273, 331, 43, 1429, 1483

Offset: 1

Vladimir Shevelev, Feb 02 2014

Also odd part of A176126(n-1) and of |A127276(n-1)|, n>=3.
Proof. By A127276 and A001788, we have odd part(A176126(n))=odd part(|A127276(n)|) = odd part(n*(n+1)-4), {odd part(A176126(n-1)), n>=3}={odd part((n+1)*(n+2)-4), n>=1}.
Let n=2^b*k, where k=k(n) is odd.
Then {odd part(A176126(n-1)), n>=3}={odd part((2^b*k+1)*(2^b*k+2)-4)}={odd part(2^(2*b)*k^2+3*2^b*k-2)}. Hence, if b>0, then {odd part(A176126(n-1), n>=3)= {odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1)}.
On the other hand, in this case odd part(a(n))=odd part(2^(b-1)*k*(2^b*k+3)-1)=odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1). It is left to consider the case of odd n. Setting n=2*m-1, m>=1, we easily find that for both expressions the odd part equals odd part(2*m^2+m-2).
The smallest prime divisor of a(n) is more than or equal to 13.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000265, A034856, A176126, A127276, A001788.

Map[#/2^IntegerExponent[#,2]&[(# (#+3)/2-1)]&,Range[100]] (* Peter J. C. Moses, Feb 02 2014 *)

a(n) = A000265(A034856(n)). - Michel Marcus, Feb 25 2025

A290031 Number of 6-cycles in the n-hypercube graph.

Original entry on oeis.org

0, 0, 0, 16, 128, 640, 2560, 8960, 28672, 86016, 245760, 675840, 1802240, 4685824, 11927552, 29818880, 73400320, 178257920, 427819008, 1016070144, 2390753280, 5578424320, 12918456320, 29712449536, 67914170368, 154350387200, 348966092800, 785173708800, 1758789107712

Offset: 0

Eric W. Weisstein, Jul 17 2017

nonn

Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Hypercube Graph.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A001788 (4-cycles).

Cf. A364688 (8-cycles).

[2^(n + 1)*Binomial(n, 3): n in [0..30]]; // Wesley Ivan Hurt, Apr 21 2021

Table[2^(n + 1) Binomial[n, 3], {n, 0, 20}]
LinearRecurrence[{8, -24, 32, -16}, {0, 0, 0, 16}, 20]
CoefficientList[Series[(16 x^3)/(-1 + 2 x)^4, {x, 0, 20}], x]
Table[Length[FindCycle[HypercubeGraph[n], {6}, All]], {n, 0, 10}] (* Eric W. Weisstein, Aug 02 2023 *)

a(n) = 2^(n + 1)*binomial(n, 3).
a(n) = 8*a(n-1)-24*a(n-2)+32*a(n-4)-16*a(n-4).
G.f.: (16*x^3)/(-1 + 2*x)^4.
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=3} 1/a(n) = 3*(2*log(2)-1)/16.
Sum_{n>=3} (-1)^(n+1)/a(n) = (3/2)^3*log(3/2) - 21/16. (End)

A345340 The number of squares with vertices from the vertices of the n-dimensional hypercube.

Original entry on oeis.org

0, 0, 1, 6, 36, 200, 1120, 6272, 35392, 200832, 1145856, 6566912, 37779456, 218050560, 1262030848, 7322034176, 42570760192, 247970693120, 1446799212544, 8453937692672, 49463868522496, 289761061240832, 1699288462655488, 9975342691254272, 58611909535989760

Offset: 0

Peter Kagey, Jun 14 2021

nonn

			For n = 4, there are a(4) = 36 such squares, nine of which contain the origin:
(0,0,0,0),(0,0,0,1),(0,0,1,0),(0,0,1,1);
(0,0,0,0),(0,0,0,1),(0,1,0,0),(0,1,0,1);
(0,0,0,0),(0,0,0,1),(1,0,0,0),(1,0,0,1);
(0,0,0,0),(0,0,1,0),(0,1,0,0),(0,1,1,0);
(0,0,0,0),(0,0,1,0),(1,0,0,0),(1,0,1,0);
(0,0,0,0),(0,1,0,0),(1,0,0,0),(1,1,0,0);
(0,0,0,0),(0,0,1,1),(1,1,0,0),(1,1,1,1);
(0,0,0,0),(0,1,0,1),(1,0,1,0),(1,1,1,1); and
(0,0,0,0),(0,1,1,0),(1,0,0,1),(1,1,1,1).

Cf. A097861, A362706.

Cf. A001788 (2-dimensional faces), A016283 (rectangles), A344854 (equilateral triangles).

a(n) = 2^(n-2) * Sum_{k=1..floor(n/2)} n!/(2*k!*k!*(n-2*k)!). - Drake Thomas, Jun 14 2021

a(n) = 2^(n-2) * A097861(n).

A084641 Binomial transform of n^7.

Original entry on oeis.org

0, 1, 130, 2574, 25904, 183200, 1040112, 5076400, 22171648, 88915968, 333209600, 1181548544, 4001402880, 13033885696, 41061830656, 125666611200, 374947708928, 1093874155520, 3128047828992, 8785866391552, 24280799641600, 66124498599936, 177683966197760

Offset: 0

Paul Barry, Jun 08 2003

The binomial transforms of n, n^2, n^3, n^4, n^5, n^6 are A001787, A001788, A058645, A058649, A059338, A056468 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-112,448,-1120,1792,-1792,1024,-256).

Cf. A001015, A001787, A001788, A058645, A058649, A059338, A056468.

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7): n in [0..40]]; // G. C. Greubel, Mar 20 2023

LinearRecurrence[{16,-112,448,-1120,1792,-1792,1024,-256}, {0,1,130, 2574,25904,183200,1040112,5076400}, 41] (* Amiram Eldar, Nov 26 2021 *)

[n^2*(n^5+21*n^4+105*n^3+35*n^2-210*n+112)*2^(n-7) for n in range(41)] # G. C. Greubel, Mar 20 2023

a(n) = n^2*(n^5 + 21*n^4 + 105*n^3 + 35*n^2 - 210*n + 112)*2^(n-7).
a(n) = Sum_{k=0..n} C(n, k)*k^7.
G.f.: x*(1+114*x+606*x^2-1168*x^3-96*x^4+816*x^5-272*x^6)/(1-2*x)^8. - Colin Barker, Sep 20 2012

A096137 Table read by rows: row n contains the sum of each nonempty subset of {1, 2, ..., n}. In each row, the sums are arranged in ascending order.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 6, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 14, 15, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12

Offset: 1

Amarnath Murthy, Jul 06 2004

The n-th row has 2^n-1 members. A001788 gives the row sums. The sums of the k-element subsets of {1, 2, ..., n} add up to A094305(n-1, k-1).

			The nonempty subsets of {1, 2, 3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}, which have sums 1, 2, 3, 3, 4, 5 and 6 respectively, so these are the terms of row 3.
Triangle T(n,k) begins:
  1;
  1, 2, 3;
  1, 2, 3, 3, 4, 5, 6;
  1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10;
  ...

Alois P. Heinz, Rows n = 1..15, flattened

Cf. A001788, A094305.

T:= proc(n) option remember; `if`(n=0, [][], subsop(1=[][],
      sort(map(x-> (x, x+n), [0, T(n-1)])))[])
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, Jul 24 2019

T[n_] := T[n] = Total /@ Subsets[Range[n], {1, n}] // Sort;
Array[T, 7] // Flatten (* Jean-François Alcover, Feb 14 2021 *)

Edited and extended by David Wasserman, Oct 04 2007

A144258 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows: T(n,k) is the number of forests of trees on n or fewer nodes using a subset of labels 1..n and k edges.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 8, 6, 3, 0, 16, 24, 27, 16, 0, 32, 80, 150, 190, 125, 0, 64, 240, 660, 1335, 1830, 1296, 0, 128, 672, 2520, 7210, 15435, 22449, 16807, 0, 256, 1792, 8736, 33040, 98105, 219912, 335160, 262144, 0, 512, 4608, 28224, 135072, 521010, 1600452, 3727962, 5902236, 4782969, 0

Offset: 0

Alois P. Heinz, Sep 16 2008

			T(3,1) = 6, because there are 6 forests of trees on 3 or fewer nodes using a subset of labels 1,2,3 and 1 edge:
  .1-2. .1... ...2. .1-2. .1.2. .1.2.
  ..... .|... ../.. ..... .|... ../..
  ..... .3... .3... .3... .3... .3...
Triangle begins:
   1;
   2,  0;
   4,  1,   0;
   8,  6,   3,   0;
  16, 24,  27,  16,   0;
  32, 80, 150, 190, 125,  0;

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to trees

Columns k = 0, 1 give A000079, A001788.
First lower diagonal gives A000272(k+1) with initial term 2.
Row sums give A144259.
Cf. A007318, A000142.

T:= proc(n, k) option remember;
      if k=0 then 2^n
    elif k<0 or n<=k then 0
    elif k=n-1 then n^(n-2)
    else add(binomial(n-1, j) *T(j+1, j) *T(n-1-j, k-j), j=0..k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..11);

T[n_, k_] := T[n, k] = Which[k == 0, 2^n, k < 0 || n <= k, 0, k == n-1, n^(n-2), True, Sum[Binomial[n-1, j]*T[j+1, j]*T[n-1-j, k-j], {j, 0, k}]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2014, translated from Maple *)

T(n,0) = 2^n, T(n,k) = 0 if k < 0 or n <= k, otherwise T(n,k) = n^(n-2) if k=n-1, otherwise T(n,k) = Sum_{j=0..k} C(n-1,j)*T(j+1,j)*T(n-1-j,k-j).

A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 4, 1, 0, 8, 12, 6, 1, 0, 16, 32, 24, 8, 1, 0, 32, 80, 80, 40, 10, 1, 0, 64, 192, 240, 160, 60, 12, 1, 0, 128, 448, 672, 560, 280, 84, 14, 1, 0, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 0, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 0

Offset: 0

Philippe Deléham, Dec 14 2011

A071919*A007318 as infinite lower triangular matrices.
A129186*A038207 as infinite lower triangular matrices.
From Paul Curtz, Nov 12 2019: (Start)
If a new main diagonal of 0's is added to the triangle, then for this variant the following propositions hold:
   The first column is A166444.
   The second column is A139756.
   The antidiagonal sums are A000129 (Pell numbers).
   The row sums are (-1)^n*A141413.
   The signed row sums are 0 followed by 1's, autosequence companion to A054977.
(End)

			Triangle begins:
   1;
   1,  0;
   2,  1,  0;
   4,  4,  1,  0;
   8, 12,  6,  1,  0;
  16, 32, 24,  8,  1, 0;
  32, 80, 80, 40, 10, 1, 0;
  ...

Essentially the same as A038207, A062715, A065109.
Cf. A001787, A001788, A139756, A000129 (antidiagonals sums).
Cf. A141413, A166444, A054977.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n<2, 1-k, 2*T(n-1, k) +T(n-1, k-1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Nov 08 2019

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2 T[n-1, k] + T[n-1, k-1];
T[0, 0] = T[1, 0] = 1; T[1, 1] = 0; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 08 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0)=T(1,0)=1 and T(1,1)=0 .
G.f.: (1-(1+y)*x)/(1-(2+y)*x).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A019590(n+1), A000012(n), A011782(n), A133494(n) for x = -2, -1, 0, 1 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A133494(n), A020699(n) for x = 0, 1, 2 respectively.
T(2n,n) = A069720(n).

A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).

Original entry on oeis.org

1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

First difference yields A295288.
1 and 7 are the only odd terms.
a(n) gives the number of words of length n defined over the alphabet {a,b,c,d} such that letters from {a,b} are only used in pairs of at most one, and consist of (a,a), (a,b) and (b,a).

			a(4) = 88. The corresponding words are cccc, cccd, ccdc, ccdd, cdcc, cdcd, cddc, cddd, dccc, dccd, dcdc, dcdd, ddcc, ddcd, dddc, dddd, caac, caca, ccaa, caad, cada, caad, cabc, cacb, ccab, cabd, cadb, cabd, cbac, cbca, ccba, cbad, cbda, cbad, daac, daca, dcaa, daad, dada, daad, dabc, dacb, dcab, dabd, dadb, dabd, dbac, dbca, dcba, dbad, dbda, dbad, aacc, acac, acca, aacd, acad, acda, aadc, adac, adca, aadd, adad, adda, abcc, acbc, accb, abcd, acbd, acdb, abdc, adbc, adcb, abdd, adbd, addb, bacc, bcac, bcca, bacd, bcad, bcda, badc, bdac, bdca, badd, bdad, bdda.

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hermann Gruber, Jonathan Lee and Jeffrey Shallit, Enumerating regular expressions and their languages, arXiv preprint arXiv:1204.4982 [cs.FL], 2012.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.

List([0..30],n->(3*n^2-3*n+8)*2^(n-3)); # Muniru A Asiru, Mar 09 2018

[(3*n^2-3*n+8)*2^(n-3): n in [0..30]]; // Vincenzo Librandi, Mar 10 2018

A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
seq(A(n), n = 0 .. 70);

Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
LinearRecurrence[{6, -12, 8}, {1, 2, 7}, 30] (* Robert G. Wilson v, Mar 07 2018 *)

makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);

a(n) = (3*n^2-3*n+8)*2^(n-3); \\ Altug Alkan, Mar 09 2018

G.f.:  (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
a(n) = 3*A001788(n) + A000079(n).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), for n >= 3, with a(0) = 1, a(1) = 2 and a(2) = 7.
a(n) = A300184(n,2).

A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 1, 16, 4, 32, 12, 64, 32, 1, 128, 80, 6, 256, 192, 24, 512, 448, 80, 1, 1024, 1024, 240, 8, 2048, 2304, 672, 40, 4096, 5120, 1792, 160, 1, 8192, 11264, 4608, 560, 10, 16384, 24576, 11520, 1792, 60, 32768, 53248, 28160, 5376, 280, 1, 65536, 114688, 67584, 15360, 1120, 12

Offset: 0

Zagros Lalo, Jul 30 2018

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2x-x^3) are given by the sequence generated by the row sums.
The row sums give A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.205569430400..., when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8,      1;
      16,      4;
      32,     12;
      64,     32,      1;
     128,     80,      6;
     256,    192,     24;
     512,    448,     80,      1;
    1024,   1024,    240,      8;
    2048,   2304,    672,     40;
    4096,   5120,   1792,    160,     1;
    8192,  11264,   4608,    560,    10;
   16384,  24576,  11520,   1792,    60;
   32768,  53248,  28160,   5376,   280,   1;
   65536, 114688,  67584,  15360,  1120,  12;
  131072, 245760, 159744,  42240,  4032,  84;
  262144, 524288, 372736, 112640, 13440, 448, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.

Row sums give A008998, A020708.
Cf. A013609
Cf. A038207
Cf. A317494
Cf. A128099, A207538.
Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).

Flat(List([0..20],n->List([0..Int(n/3)],k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018

/* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018

t[n_, k_] := t[n, k] = 2^(n - 3k)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).

A318776 Triangle read by rows: T(0,0) = 1; T(n,k) = 2*T(n-1,k) + T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 1, 64, 4, 128, 12, 256, 32, 512, 80, 1024, 192, 1, 2048, 448, 6, 4096, 1024, 24, 8192, 2304, 80, 16384, 5120, 240, 32768, 11264, 672, 1, 65536, 24576, 1792, 8, 131072, 53248, 4608, 40, 262144, 114688, 11520, 160, 524288, 245760, 28160, 560, 1048576, 524288, 67584, 1792, 1, 2097152, 1114112, 159744, 5376, 10

Offset: 0

Zagros Lalo, Sep 04 2018

The numbers in rows of the triangle are along a "fourth layer" skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "fourth layer" skew diagonals pointing top-right in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2*x-x^5) are given by the sequence generated by the row sums.
The row sums give A098588.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.0559673967128..., when n approaches infinity.

			Triangle begins:
        1;
        2;
        4;
        8;
       16;
       32,       1;
       64,       4;
      128,      12;
      256,      32;
      512,      80;
     1024,     192,      1;
     2048,     448,      6;
     4096,    1024,     24;
     8192,    2304,     80;
    16384,    5120,    240;
    32768,   11264,    672,    1;
    65536,   24576,   1792,    8;
   131072,   53248,   4608,   40;
   262144,  114688,  11520,  160;
   524288,  245760,  28160,  560;
  1048576,  524288,  67584, 1792,  1;
  2097152, 1114112, 159744, 5376, 10;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A098588.
Cf. A013609, A038207, A128099, A207538.
Cf. also A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3)

t[n_, k_] := t[n, k] = 2^(n - 5 k)/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 21}, {k, 0, Floor[n/5]}] // Flatten

T(n,k) = 2^(n - 5*k) / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5).

cp's OEIS Frontend

A236999 Odd part of n*(n+3)/2-1 (A034856).

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A290031 Number of 6-cycles in the n-hypercube graph.

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A345340 The number of squares with vertices from the vertices of the n-dimensional hypercube.

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A084641 Binomial transform of n^7.

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A096137 Table read by rows: row n contains the sum of each nonempty subset of {1, 2, ..., n}. In each row, the sums are arranged in ascending order.

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A144258 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows: T(n,k) is the number of forests of trees on n or fewer nodes using a subset of labels 1..n and k edges.

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A183190 Triangle T(n,k), read by rows, given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).

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A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).