A002662 -id:A002662 - cp's OEIS frontend

A061290 Square array read by antidiagonals of T(n,k) = T(n-1,k) + T(n-1, floor(k/2)) with T(0,0)=1.

1, 0, 2, 0, 1, 4, 0, 0, 3, 8, 0, 0, 1, 7, 16, 0, 0, 1, 4, 15, 32, 0, 0, 0, 4, 11, 31, 64, 0, 0, 0, 1, 11, 26, 63, 128, 0, 0, 0, 1, 5, 26, 57, 127, 256, 0, 0, 0, 1, 5, 16, 57, 120, 255, 512, 0, 0, 0, 1, 5, 16, 42, 120, 247, 511, 1024, 0, 0, 0, 0, 5, 16, 42, 99, 247, 502, 1023, 2048, 0, 0

Offset: 0

Henry Bottomley, May 22 2001

Row sums give 3^n.

			T(9,3) = T(8,3) + T(8,floor(3/2)) = T(8,3) + T(8,1) = 247 + 255 = 502. Rows start (1,0,0,0,0,...), (2,1,0,0,0,...), (4,3,1,1,0,...), (8,7,4,4,1,...), etc.

Row sums are A000244. Columns are A000079, A000225, A000295 twice, A002662 four times, A002663 eight times, A002664 sixteen times, A035038 thirty two times, etc.

T(n, k) = C(n, 0) + C(n, 1) + ... + C(n, n-ceiling(log_2(k+1))) = 2^n - C(n, 0) - C(n, 1) - ... - C(n, floor(log_2(k))) = A008949(n, n-A029837(k+1)) = A000079(n) - A008949(n, A000523(k)).

A132795 Triangle of Gely numbers, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 5, 0, 1, 0, 16, 6, 1, 1, 0, 42, 56, 21, 0, 1, 0, 99, 316, 267, 36, 1, 1, 0, 219, 1408, 2367, 960, 85, 0, 1, 0, 466, 5482, 16578, 14212, 3418, 162, 1, 1, 0, 968, 19624, 99330, 153824, 77440, 11352, 341, 0, 1, 0, 1981, 66496, 534898, 1364848, 1233970, 389104, 36829, 672, 1

Offset: 0

Olivier Gérard, Aug 31 2007

First row is for n=0. First column is for k=0.
Sum of rows is n! = permutations of n symbols (A000142)
These numbers are related to the Eulerian numbers A(n,k).
Third Column (k=2) is A002662(n+1).
Second Diagonal (k=n-1) is A132796.
Binomial transform of this triangle gives set partitions without singletons (in a form very close to array A105794).

			Triangle starts:
1;
1, 0;
1, 0, 1;
1, 0, 5, 0;
1, 0, 16, 6, 1;
1, 0, 42, 56, 21, 0;
...

Charles O. Gely, Un tableau de conversion des polynomes cyclotomiques cousin des nombres Euleriens, Preprint Univ. Paris 7, 1999.
Olivier Gérard, Quelques facons originales de compter les permutations, submitted to Knuth07.
Olivier Gérard and Karol Penson, Set partitions, Multiset permutations and bi-permutations, in preparation.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, p. 269.

Cf. A000296, A132796.

T(n,k)= sum(j=0, k, (-1)^j*binomial(n+1, j)*sum(m=0, n, (k-j)^m)); \\ Michel Marcus, Jun 04 2014

T(n,k) = sum(j=0..k, (-1)^j*C(n+1,j)*sum(m=0..n, (k-j)^m) ).

A156732 Triangle T(n, k) = ((n-2k)^2/(n-k+1))binomial(n+1, k+1), read by rows.

Original entry on oeis.org

0, 1, 1, 4, 0, 4, 9, 2, 2, 9, 16, 10, 0, 10, 16, 25, 27, 5, 5, 27, 25, 36, 56, 28, 0, 28, 56, 36, 49, 100, 84, 14, 14, 84, 100, 49, 64, 162, 192, 84, 0, 84, 192, 162, 64, 81, 245, 375, 270, 42, 42, 270, 375, 245, 81

Offset: 0

Roger L. Bagula, Feb 14 2009

The general formula for integrals of the form int (x^(2*k))/((arcsin(x))^2) dx involves the triangular sequence t(2*k, n). For example, the solution to the integral Integral x^6/((arcsin(x))^2) dx involves the following sequence: -5*Si(arcsin(x)) + 27*Si(3*arcsin(x)) - 25*Si(5*arcsin(x)), where Si represents the sine integral. The sequence of integers 5, 27, 25 corresponds to the sixth row of this triangular sequence. The general formula for the integral Integral x^(2*k)/((arcsin(x))^2) dx is: (1/(2^(2*k)))*( -((2^(2*k)*sqrt(1-x^2)*(x^(2*k)))/arcsin(x) )+ (-1)^(k+1)*(1+(2*k))*Si((1+2*k)*arcsin(x)) + Sum_{n=1..k} (-1)^n*((1-2*n)^2/(k-n+1))*binomial(2*k, k+n)*Si((2*n-1)*arcsin(x)) ). - John M. Campbell, Sep 22 2010

			Triangle begins as:
   0;
   1,   1;
   4,   0,   4;
   9,   2,   2,   9;
  16,  10,   0,  10, 16;
  25,  27,   5,   5, 27, 25;
  36,  56,  28,   0, 28, 56,  36;
  49, 100,  84,  14, 14, 84, 100,  49;
  64, 162, 192,  84,  0, 84, 192, 162,  64;
  81, 245, 375, 270, 42, 42, 270, 375, 245, 81;

J. Riordan, Combinatorial Identities, Wiley, 1968.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
John. M. Campbell, Double Series Involving Binomial Coefficients and the Sine Integral, arXiv:1009.0236 [math.NT], 2010, p. 3-4. [From _John M. Campbell_, Sep 22 2010]

Cf. A000290, A002662.

A156732:= func< n,k | ((n-2*k)^2/(n+2))*Binomial(n+2, k+1) >;
[A156732(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 28 2021

T[n_, k_]:= ((n-2*k)^2/(n-k+1))*Binomial[n+1, k+1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 28 2021 *)

def A156732(n, k): return ((n-2*k)^2/(n+2))*binomial(n+2, k+1)
flatten([[A156732(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2021

T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1).
Sum_{k=0..floor(n/2)} T(n-1, k-1) = 2^n.
From G. C. Greubel, Feb 28 2021: (Start)
T(n, k) = T(n, n-k).
T(n, k) = ((n-2*k)^2/(n+2))*binomial(n+2, k+1).
Sum_{k=0..n} T(n, k) = 2*(2^(n+1) -n-2) = 4*A002662(n) + 2*n^2. (End)

Edited by G. C. Greubel, Feb 28 2021

A193590 Augmentation of the Euler triangle A008292. See Comments.

Original entry on oeis.org

1, 1, 1, 1, 5, 2, 1, 16, 33, 8, 1, 42, 275, 342, 58, 1, 99, 1669, 6441, 5600, 718, 1, 219, 8503, 82149, 217694, 143126, 14528, 1, 466, 39076, 843268, 5466197, 10792622, 5628738, 466220, 1, 968, 168786, 7621160, 107506633, 509354984, 788338180

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193590, (column 1)=A002662, with general term 2^n-1-n(n+1)/2.

			First 5 rows of A193589:
1
1....1
1....5....2
1....16...33....8
1....42...275...342....58

Cf. A008292, A193091.

p[n_, k_] :=
Sum[((-1)^j)*((k + 1 - j)^(n + 1))*Binomial[n + 2, j], {j, 0, k + 1}]
(* A008292, Euler triangle *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]]  (* A193590  *)
Flatten[Table[v[n], {n, 0, 8}]]

A229611 Expansion of 1/((1-x)^3*(1-11x)).

Original entry on oeis.org

1, 14, 160, 1770, 19485, 214356, 2357944, 25937420, 285311665, 3138428370, 34522712136, 379749833574, 4177248169405, 45949729863560, 505447028499280, 5559917313492216, 61159090448414529, 672749994932559990, 7400249944258160080, 81402749386839761090

Offset: 0

Yahia Kahloune, Sep 26 2013

This sequence was chosen to illustrate a method of matching generating functions and closed-form solutions: The general term associated with the generating function 1/((1-s*x)^3*(1-r*x)) with r>s>=1 is a(n) = [r^(n+3) - s^(n+1)*(s^2 + (r-s)*s*binomial(n+3,1) +(r-s)^2*binomial(n+3,2))] / (r-s)^3 .

			a(3) = (11^6 - (50*3^2+260*3 + 331))/1000 = 1770 .

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-36,34,-11).

Cf. A002662, A052150, A052161, A052244, A052262.

[(11^(n+3) - (50*n^2 + 260*n + 331))/1000: n in [0..25]]; // Vincenzo Librandi, Sep 27 2013

CoefficientList[Series[1/((1 - x)^3 (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 27 2013 *)
LinearRecurrence[{14,-36,34,-11},{1,14,160,1770},30] (* Harvey P. Dale, Apr 09 2016 *)

a(n) = (11^(n+3) - (1 + 10*C(n+3,1) + 100*C(n+3,2)))/1000 = (11^(n+3) - (50*n^2 + 260*n + 331))/1000.

a(n) = 14*a(n-1) -36*a(n-2) +34*a(n-3) -11*a(n-4). - Vincenzo Librandi, Sep 27 2013

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

A326292 Number of crossing integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 80, 105, 142, 186, 248, 320, 421, 539, 698, 889, 1140, 1438, 1827, 2291, 2882, 3593, 4489, 5559, 6902, 8503, 10484, 12853, 15763

Offset: 0

Gus Wiseman, Oct 03 2019

nonn

A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. An integer partition is crossing if, by replacing each part with its multiset of prime indices, we obtain a crossing multiset partition.

			The a(31) = 1 through a(36) = 7 partitions:
  21,10  21,10,1  21,10,2    21,10,3      21,10,4        21,10,5
                  21,10,1,1  21,10,2,1    21,10,2,2      21,10,3,2
                             21,10,1,1,1  21,10,3,1      21,10,4,1
                                          21,10,2,1,1    21,10,2,2,1
                                          21,10,1,1,1,1  21,10,3,1,1
                                                         21,10,2,1,1,1
                                                         21,10,1,1,1,1,1

The Heinz numbers of these partitions are given by A324170.

Cf. A000108, A002662, A016098, A056239, A112798, A324172, A324326, A326210, A326252, A326332.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

More terms from Jinyuan Wang, Jun 28 2020

A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

Original entry on oeis.org

2, 3, -2, 4, -5, 2, 5, -9, 7, -2, 6, -14, 16, -9, 2, 7, -20, 30, -25, 11, -2, 8, -27, 50, -55, 36, -13, 2, 9, -35, 77, -105, 91, -49, 15, -2, 10, -44, 112, -182, 196, -140, 64, -17, 2, 11, -54, 156, -294, 378, -336, 204, -81, 19, -2

Offset: 1

Clark Kimberling, Sep 24 2022

n-th row sum = 1 for n >= 2.

			First 7 rows:
  2
  3      -2
  4      -5       2
  5      -9       7     -2
  6     -14      16     -9     2
  7     -20      30    -25    11     -2
  8     -27      50    -55    36    -13     2
Row 4 gives recurrence coefficients for the sequence
(r(k)) = (A002662(k)) = (0,0,0,1,5,16,42,99,219,...); i.e.,
r(k) = 5*r(k-1) - 9*r(k-2) + 7*r(k-3) - 2*r(k-4),
with initial values (r(0), r(1), r(2), r(3)) = (0,0,0,1).
(Here r(k) = number of subsets of {1,2,...,4} having at least 3 elements.)

Cf. A029638, A029635.

Cf. sequences generated by recurrences, by row, beginning with row 1: A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039.

Table[Binomial[n, k]*(-1)^(k - 1)*(n + k)/n, {n, 1, 12}, {k, 1, n}]

T(n,k) = (-1)^(k-1) * (C(n,k) + C(n-1,k-1)), for n >= 1, k >= 1.

T(n,k) = (-1)^(k-1) * C(n,k)*(n+k)/n, for n >= 1, k >= 1.

cp's OEIS Frontend

A061290 Square array read by antidiagonals of T(n,k) = T(n-1,k) + T(n-1, floor(k/2)) with T(0,0)=1.

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A132795 Triangle of Gely numbers, read by rows.

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A156732 Triangle T(n, k) = ((n-2*k)^2/(n-k+1))*binomial(n+1, k+1), read by rows.

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A193590 Augmentation of the Euler triangle A008292. See Comments.

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A229611 Expansion of 1/((1-x)^3*(1-11x)).

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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A326292 Number of crossing integer partitions of n.

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A357255 Triangular array: row n gives the recurrence coefficients for the sequence (c(k) = number of subsets of {1,2,...,n} that have at least k-1 elements) for k >= 1.

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A156732 Triangle T(n, k) = ((n-2k)^2/(n-k+1))binomial(n+1, k+1), read by rows.