A002697 -id:A002697 - cp's OEIS frontend

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1

Offset: 0

Philippe Deléham, Jan 25 2004

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Corrected by Alford Arnold, Oct 18 2006

A133789 Let P(A) denote the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.

Original entry on oeis.org

0, 1, 4, 16, 70, 316, 1414, 6196, 26590, 112156, 466774, 1923076, 7863310, 31972396, 129459334, 522571156, 2104535230, 8460991036, 33972711094, 136277478436, 546270602350, 2188566048076, 8764718254054, 35090241492916, 140455083984670, 562102715143516

Offset: 0

Ross La Haye, Jan 03 2008, Jan 08 2008

nonn

Also, number of even binomial coefficient in rows 0 to 2^n of Pascal's triangle. [Aaron Meyerowitz, Oct 29 2013]

			a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
{1} and {2},
{1} and {3},
{2} and {3},
{1} and {2,3},
{2} and {1,3},
{3} and {1,2}
are disjoint, while
{} and {1},
{} and {2},
{} and {3},
{} and {1,2},
{} and {1,3},
{} and {2,3},
{} and {1,2,3}
are disjoint and one is a subset of the other and
{1,2} and {1,3},
{1,2} and {2,3},
{1,3} and {2,3}
are intersecting, but neither is a subset of the other.
Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then  6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [_Ross La Haye_, Feb 22 2009]

Cf. A082134, A002697, A020522, A006516, A007582, A000302.

Cf. A000225, A000392, A032263.

a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - R. J. Mathar, Jan 11 2008
a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - Vladeta Jovovic, Jan 04 2008, corrected by Eric Rowland, May 15 2017
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [Aaron Meyerowitz, Oct 29 2013]

Edited by N. J. A. Sloane, Jan 20 2008 to incorporate suggestions from several contributors.

A228310 The hyper-Wiener index of the hypercube graph Q(n) (n>=2).

Original entry on oeis.org

10, 72, 448, 2560, 13824, 71680, 360448, 1769472, 8519680, 40370176, 188743680, 872415232, 3992977408, 18119393280, 81604378624, 365072220160, 1623497637888, 7181185318912, 31610959298560, 138538465099776, 604731395276800, 2630031813640192

Offset: 2

Emeric Deutsch, Aug 20 2013

The hypercube graph Q(n) has as vertices the binary words of length n and an edge joins two vertices whenever the corresponding binary words differ in just one place.

Q(n) is distance-transitive and therefore also distance-regular. The intersection array is {n,n-1,n-2,...,1; 1,2,3,...,n-1,n}.

Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993 (p. 161).

R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
Eric Weisstein's World of Mathematics, Hypercube Graph.
Index entries for linear recurrences with constant coefficients, signature (12,-48,64)

Cf. A143376, A002697

a := proc (n) options operator, arrow: 4^(n-2)*n*(3+n) end proc: seq(a(n), n = 2 .. 25);

LinearRecurrence[{12,-48,64},{10,72,448},30] (* Harvey P. Dale, Dec 13 2024 *)

a(n) = 4^{n-2}*n*(3+n).
G.f.: 2*x^2*(5 - 24*x + 32*x^2)/(1-4*x)^3.
The Hosoya-Wiener polynomial of Q(n) is 2^{n-1}*((1+t)^n - 1).

A254630 Ascending antidiagonal numerators of the table of repeated differences of A164558(n)/A027642(n).

Original entry on oeis.org

1, 1, 3, 1, 2, 13, 0, 1, 5, 3, -1, -1, 2, 29, 119, 0, -1, -1, 1, 31, 5, 1, 1, -1, -8, -1, 43, 253, 0, 1, 1, 4, -4, -1, 41, 7, -1, -1, -1, 4, 8, 4, -1, 29, 239, 0, -1, -1, -8, -4, 4, 8, 1, 31, 9, 5, 5, 7, -4, -116, -32, -116, -4, 7, 71, 665, 0

Offset: 0

Paul Curtz, Feb 03 2015

The difference table of Bernoulli(n,2) or B(n,2) = A164558(n)/A027642(n) is defined by placing the fractions in the upper row and calculating further rows as the differences of their preceding row:
1,       3/2, 13/6,     3, 119/30, ...
1/2,     2/3,  5/6, 29/30, ...
1/6,     1/6, 2/15, ...
0,     -1/30, ...
-1/30, ...
etc.
The first column is A164555(n)/A027642(n).
In particular, the sums of the antidiagonals
1                              = 1
1/2 + 3/2                      = 2
1/6 + 2/3 + 13/6               = 3
0   + 1/6 +  5/6 + 3           = 4
etc. are the positive natural numbers. (This is rewritten for Bernoulli(n,3) in A157809).
We also have for Bernoulli(.,2)
B(0,2) = 1
B(0,2) + 2*B(1,2) = 4
B(0,2) + 3*B(1,2) + 3*B(2,2) = 12
B(0,2) + 4*B(1,2) + 6*B(2,2) + 4*B(3,2) = 32
etc. with right hand sides provided by A001787.
More generally sum_{s=0..t-1} binomial(t,s)*Bernoulli(s,q) gives A027471(t) for q=3, A002697 for q=4 etc, by reading A104002 downwards the q-th column.

Cf. A027641, A027642, A074909, A085737, A085738, A104002, A157809, A157920, A157930, A157945, A157946, A157965, A164555, A164558, A190339, A158302, A181131 (numerators and denominators of the main diagonal).

nmax = 11; A164558 = Table[BernoulliB[n,2], {n, 0, nmax}]; D164558 = Table[ Differences[A164558, n], {n, 0, nmax}]; Table[ D164558[[n-k+1, k+1]] // Numerator, {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2015 *)

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 12, 6, 1, 0, 0, 5, 32, 27, 8, 1, 0, 0, 6, 80, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (2*k, -k^2).

			Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Antidiagonal sums: A101495.
Column 1 is column 2 of A300453.
Column 2 is column 1 of A300184.
Cf. A104002, A320530.

T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten

T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
G.f. for columns: x/(1 - k*x)^2.
E.g.f. for columns: x*exp(k*x).
T(n,1) = A001477(n).
T(n,2) = A001787(n).
T(n,3) = A027471(n+1).
T(n,4) = A002697(n).
T(n,5) = A053464(n).
T(n,6) = A053469(n), n > 0.
T(n,7) = A027473(n), n > 0.
T(n,8) = A053539(n).
T(n,9) = A053540(n), n > 0.
T(n,10) = A053541(n), n > 0.
T(n,11) = A081127(n).
T(n,12) = A081128(n).

A335087 Row sums of A335436.

Original entry on oeis.org

1, 7, 34, 150, 628, 2540, 10024, 38840, 148368, 560368, 2096928, 7786592, 28726592, 105390272, 384788096, 1398978432, 5067403520, 18294707968, 65854095872, 236421150208, 846732997632, 3025927678976, 10792083499008, 38420157773824, 136547503083520, 484546494459904, 1716976084393984

Offset: 0

Oboifeng Dira, Sep 11 2020

nonn

This sequence is also a composition of generating functions H(x) = G(F(x)), where G(x) = x/(1-4*x)^2 is the generating function of A002697 and F(x) = x*(1-x)/(1-2*x^2) is the generating function of 0, A016116*(-1)^n.

			For n = 4, a(4) = 8*a(3)-20*a(2)+16*a(1)-4*a(0) = 8*150-20*34+16*7-4*1 = 628.

Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-4).

Composition of g.fs of A002697 and A016116.

Cf. A335436.

f:=x->x*(1-x)/(1-2*x^2):g:=x->(x)/(1-4*x)^2:
C:=n->coeff(series(g(f(x))/x,x,n+1),x,n): seq(C(n),n=0..30);

a(n) = 8*a(n-1)-20*a(n-2)+16*a(n-3)-4*a(n-4), a(0)=1, a(1)=7, a(2)=34, a(3)=150 for  n>=4.
G.f.: (1-x)*(1-2*x^2)/(1-4*x+2*x^2)^2.
a(0)=1; a(n) = 2*n+1+Sum_{k=1..n}[(2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1)]*(2n-k+1)/(4*sqrt(2)), n>=1.
G.f.: G(F(x))/x where G(x) is g.f of A002697 and F(x) is g.f of 0,A016116*(-1)^n.

A339240 a(n) = n2^(2n-2) + nbinomial(2n,n)/2.

Original entry on oeis.org

0, 2, 14, 78, 396, 1910, 8916, 40684, 182552, 808614, 3545220, 15414212, 66556584, 285707708, 1220340296, 5189913240, 21988512304, 92850096902, 390913863012, 1641450064084, 6876023427080, 28741451864916, 119902111845208, 499304732388968, 2075821104461136, 8617006998238300

Offset: 0

Michel Marcus, Nov 28 2020

nonn

Horst Alzer and Helmut Prodinger, Identities and Inequalities for Sums Involving Binomial Coefficients, INTEGERS 20 (2020) A9.

Cf. A002697, A002457.

a[n_] := n*(2^(2*n - 2) + Binomial[2*n, n]/2); Array[a, 26, 0] (* Amiram Eldar, Nov 28 2020 *)

a(n) = n*2^(2*n-2) + n*binomial(2*n,n)/2;

a(n) = sum(k=0, n, binomial(n,k)*k*sum(j=0, k, binomial(n, j)));

a(n) = Sum_{k=0..n} binomial(n, k)*k*Sum_{j=0..k} binomial(n, j).
a(n) = A002697(n) + A002457(n-1), for n>0.
G.f.: x*(1/(1 - 4*x)^2 + 1/(1 - 4*x)^(3/2)). - Stefano Spezia, Nov 28 2020

A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 5, 0, 4, 2, 10, 1, 7, 5, 19, 0, 8, 4, 20, 2, 14, 10, 38, 1, 11, 7, 29, 5, 23, 19, 65, 0, 16, 8, 40, 4, 28, 20, 76, 2, 22, 14, 58, 10, 46, 38, 130, 1, 19, 11, 49, 7, 37, 29, 103, 5, 31, 23, 85, 19, 73, 65, 211

Offset: 0

Ryan Brooks, Oct 04 2021

			n\k 0  1  2  3  4  5  6  7
0   0
1   0  1
2   0  2  1  5
3   0  4  2 10  1  7  5 19

Cf. A001047 (right diagonal), A002697 (row sums), A119733.

Cf. A133457 (binary exponents).

T[0, 0] = 0; T[n_, k_] := T[n, k] = If[EvenQ[k], T[n - 1, k/2], 3*T[n - 1, (k - 1)/2] + 2^(n - 1)]; Table[T[n, k], {n, 0, 5}, {k, 0, 2^n - 1}] // Flatten (* Amiram Eldar, Oct 11 2021 *)

T(n, k) = if ((n==0) && (k==0), 0, if (k%2, 3*T(n-1,(k-1)/2) + 2^(n-1), T(n-1,k/2)));
tabf(nn) = for (n=0, nn, for (k=0, 2^n-1, print1(T(n,k), ", ")); print); \\ Michel Marcus, Oct 18 2021

T(n,k) = my(ret=0); for(i=0,n-1, if(bittest(k,n-1-i), ret=3*ret+1<Kevin Ryde, Oct 22 2021

T(n,k) = T(n-1,k/2) for k being even.
T(n,k) = 3*T(n-1,(k-1)/2) + 2^(n-1) for k being odd.
T(n,k) = 2*T(n-1,k) for 0 <= k <= 2^(n-1) - 1.
T(n,k) = Sum_{i=0..r} 2^(n-1-e[i]) * 3^i where binary expansion k = 2^e[0] + 2^e[1] + ... + 2^e[r] with ascending e[0] < e[1] < ... < e[r] (A133457). - Kevin Ryde, Oct 22 2021

A378504 Expansion of (Sum_{k>=0} binomial(3k,k) x^k)^4.

Original entry on oeis.org

1, 12, 114, 984, 8055, 63744, 492702, 3742704, 28053423, 208057260, 1529802648, 11168142048, 81041199876, 585045970992, 4204705925670, 30101448952032, 214756404746031, 1527491122906212, 10834911076417458, 76666402505673720, 541277205506059743

Offset: 0

Seiichi Manyama, Nov 28 2024

nonn

Marko Riedel et al, Identity between two generating functions

Cf. A005809, A006256, A378483.

Cf. A002697, A378484.

nmax = 20; CoefficientList[Series[Sum[Binomial[3*k, k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^4)

a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k) * binomial(3*l,l).
G.f.: B(x)^4 where B(x) is the g.f. of A005809.
4*a(n) - 27*a(n-1) = 3*A006256(n) + A005809(n) for n > 0.
Sum_{n >= 0} a(n) * z^n / (1+z)^(3*n) = (1+z)^4 / (1-2*z)^4. - Marko Riedel, Jul 22 2025
From Vaclav Kotesovec, Jul 23 2025: (Start)
Recurrence: 8*(n-1)*n*(2*n - 1)*a(n) = 6*(n-1)*(36*n^2 - 9*n - 5)*a(n-1) - 81*n*(3*n - 2)*(3*n - 1)*a(n-2).
a(n) ~ n * 3^(3*n+2) / 2^(2*n+4). (End)

cp's OEIS Frontend

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

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A228310 The hyper-Wiener index of the hypercube graph Q(n) (n>=2).

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A254630 Ascending antidiagonal numerators of the table of repeated differences of A164558(n)/A027642(n).

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A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

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A335087 Row sums of A335436.

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A339240 a(n) = n*2^(2*n-2) + n*binomial(2*n,n)/2.

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A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

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A339240 a(n) = n2^(2n-2) + nbinomial(2n,n)/2.

A378504 Expansion of (Sum_{k>=0} binomial(3k,k) x^k)^4.