A074909 -id:A074909 - cp's OEIS frontend

A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

1, 1, 1, 0, 2, 1, -2, 2, 3, 1, -5, 0, 5, 4, 1, -9, -5, 5, 9, 5, 1, -14, -14, 0, 14, 14, 6, 1, -20, -28, -14, 14, 28, 20, 7, 1, -27, -48, -42, 0, 42, 48, 27, 8, 1, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -44, -110, -165, -132, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Gary W. Adamson, Jun 14 2007

Row sums = n.

A131085 * A000012 = A074909 starting (1, 2, 1, 3, 3, ...) instead of (1, 1, 2, 1, 3, 3, ...).

			First few rows of the triangle are:
   1;
   1,  1;
   0,  2, 1;
  -2,  2, 3, 1;
  -5,  0, 5, 4, 1;
  -9, -5, 5, 9, 5, 1;
-14, -14, 0, 14, 14, 6, 1;
-20, -28, -14, 14, 28, 20, 7, 1;
-27, -48, -42, 0, 42, 48, 27, 8, 1;
-35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
   ...

Cf. A007318, A009766, A097808, A112466, A214292, A112467, A129686, A131084.

tabl(nn) = {t007318 = matrix(nn, nn, n, k, binomial(n-1, k-1)); t129686 = matrix(nn, nn, n, k, (k<=n)*((-1)^((n-k)\2)*((k==n) || (-1)*(k==(n-2))))); t131085 = t007318*t129686; for (n = 1, nn, for (k = 1, n, print1(t131085[n, k], ", ");););} \\ Michel Marcus, Feb 12 2014

Binomial transform of A129686 signed with (1, 1, 1, ...) in the main diagonal and (-1, -1, -1, ...) in the subsubdiagonal.
T(n,m) = T(n-1,m-1) + T(n-1,m). - Yuchun Ji, Dec 17 2018
T(2*k,k-1) = 0 for k > 0. - Yuchun Ji, Dec 20 2018
Comparing this triangle with the Catalan triangle A009766 one can see many similarities. For example, T(k+j,k) = A009766(k+1,j) for j < k+2. - Yuchun Ji, Dec 23 2018 [Edited by N. J. A. Sloane, Feb 11 2019]

Missing comma corrected by Naruto Canada, Aug 26 2007
More terms from Michel Marcus, Feb 12 2014
Offset changed by N. J. A. Sloane, Feb 11 2019

A193465 Row sums of triangle A061312.

Original entry on oeis.org

0, 2, 9, 52, 335, 2466, 20447, 189064, 1930959, 21603430, 262869959, 3457226268, 48880169351, 739429561066, 11918051268255, 203914545928336, 3691384616598047, 70491995143458894, 1416242276574905879, 29862732908481855460, 659413025994777460119

Offset: 0

Johannes W. Meijer, Jul 27 2011

a(n) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A001563(k) for k = 0, 1, ..., n. - Michael Somos, Jun 06 2012

			2*x + 9*x^2 + 52*x^3 + 335*x^4 + 2466*x^5 + 20447*x^6 + 189064*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A000166, A001563, A002467, A047920, A061312, A180191, A074909.

A193465 := proc(n): add(A061312(n,k), k=0..n) end: A061312:=proc(n,k): add(((-1)^j)*binomial(k+1,j)*(n+1-j)!, j=0..k+1) end: seq(A193465(n), n=0..20);

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 + x - (1 + x^2) / Exp[ x ]) / (1 - x)^3, {x, 0, n}]] (* Michael Somos, Jun 06 2012 *)

{a(n) = if( n<0, 0, n! * polcoeff( (1 + x - (1 + x^2) / exp(x + x * O(x^n))) / (1 - x)^3, n))} /* Michael Somos, Jun 06 2012 */

a(n) = Sum_{k=0..n} A061312(n,k).
a(n) = (n+1)*A180191(n+1).
a(n) = A002467(n+2) - (n+1)! (the game of mousetrap with n cards).
a(n) = (n+1)*(n+1)! - A000166(n+2) (rencontres numbers).
a(n) = ((n-n^3)*a(n-3) + (2*n+n^2-n^3)*a(n-2) - (1-n-2*n^2)*a(n-1))/n with a(0) = 0, a(1) = 2 and a(2) = 9.
E.g.f: (1 + x - (1 + x^2) / exp(x)) / (1 - x)^3. - Michael Somos, Jun 06 2012
a(n) = Sum_{k=0..n} C(n+1,k)*A000166(k+1) = Sum_{k=0..n} A074909(n,k)*A000166(k+1). - Anton Zakharov, Sep 26 2016
a(n) = Sum_{k=1..n+1} A047920(n+1,k). - Alois P. Heinz, Sep 01 2021

A071761 Dealing cards in a game of solitaire.

Original entry on oeis.org

0, 1, 2, 21, 31, 32, 321, 421, 431, 432, 4321, 5321, 5421, 5431, 5432, 54321, 64321, 65321, 65421, 65431, 65432, 654321, 754321, 764321, 765321, 765421, 765431, 765432, 7654321, 8654321, 8754321, 8764321, 8765321, 8765421, 8765431, 8765432

Offset: 0

Allan C. Wechsler, Jun 07 2002

An infinite square array,
a11 a12 a13 ...
a21 a22 a23 ...
a31 a32 a33 ...
a41 a42 a43 ...
...............
is built up by antidiagonals: a11 a21 a12 a31 a22 a13 ... The terms of the sequence record the number of entries in each column at each step, e.g. when dealing cards in a game of solitaire.

The parity of this sequence gives A023532.

Cf. A071762, A074909.

Entry revised by Jeremy Gardiner, Dec 08 2004

A168557 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial (-1)^n*((x + 1)^n - x^n + 1), 0 <= k <= max(0, n - 1).

Original entry on oeis.org

1, -2, 2, 2, -2, -3, -3, 2, 4, 6, 4, -2, -5, -10, -10, -5, 2, 6, 15, 20, 15, 6, -2, -7, -21, -35, -35, -21, -7, 2, 8, 28, 56, 70, 56, 28, 8, -2, -9, -36, -84, -126, -126, -84, -36, -9, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, -2, -11, -55, -165, -330, -462, -462, -330

Offset: 0

Roger L. Bagula, Nov 29 2009

A variant of Pascal's triangle, the first column replaced by 2 (if n > 0), the last column dropped, and then odd rows multiplied by (-1)^n.

Absolute value row sums are A000079.

			Triangle begins:
   1;
  -2;
   2,   2;
  -2,  -3,  -3;
   2,   4,   6,    4;
  -2,  -5, -10,  -10,   -5;
   2,   6,  15,   20,   15,    6;
  -2,  -7, -21,  -35,  -35,  -21,   -7;
   2,   8,  28,   56,   70,   56,   28,    8;
  -2,  -9, -36,  -84, -126, -126,  -84,  -36,   -9;
   2,  10,  45,  120,  210,  252,  210,  120,   45,  10;
  -2, -11, -55, -165, -330, -462, -462, -330, -165, -55, -11;
   2,  12,  66,  220,  495,  792,  924,  792,  495,  220, 66, 12;
   ...

Cf. A074909, A007318, A108086, A117440, A130595.

Table[CoefficientList[(-1)^n*(x + 1)^n - (-1)^n*(x^n - 1), x], {n, 0, 12}]

create_list((-1)^n*binomial(n, k) + (-1)^n*kron_delta(0, k) - kron_delta(0, n), n, 0, 12, k, 0, max(0, n - 1)); /* Franck Maminirina Ramaharo, Nov 21 2018 */

From Franck Maminirina Ramaharo, Nov 22 2018: (Start)
T(n,k) = (-1)^n*binomial(n, k) + (-1)^n*delta(0, k) - delta(0, n), where delta is Kronecker's delta-symbol.
G.f.: (1 + 2*x*y - (1 - x - x^2)*y^2)/((1 + y)*(1 + x*y)*(1 + y + x*y)).
E.g.f.: (1 - exp(y) + exp(x*y))*exp(-(1 + x)*y). (End)

A199011 Triangle T(n,k), read by rows, given by (1,1,-1,1,0,0,0,0,0,0,0,...) DELTA (0,1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 1, 0, 7, 21, 35, 35, 21, 7, 1, 0, 8, 28, 56, 70, 56, 28, 8, 1, 0, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0

Offset: 0

Philippe Deléham, Nov 01 2011

Mirror image of triangle in A198321.

Variant of A074909, A135278.

			Triangle begins :
1
1, 0
2, 1, 0
3, 3, 1, 0
4, 6, 4, 1, 0
5, 10, 10, 5, 1, 0
6, 15, 20, 15, 6, 1, 0

Cf. A007318, A074909, A135278, A198321

T(n,k)=binomial(n,k+1).
Sum_{0<=k<=n} T(n,k)*x^k = ((x+1)^n-1)/x for n>0.
G.f.: (1-(1+y)*x+(1+y)*x^2)/(1-(2+y)*x+(1+y)*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

A209518 Triangle by rows, reversal of A104712.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 15, 20, 15, 1, 7, 21, 35, 35, 21, 1, 8, 28, 56, 70, 56, 28, 1, 9, 36, 84, 126, 126, 84, 36, 1, 10, 45, 120, 210, 252, 210, 120, 45, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55

Offset: 0

Gary W. Adamson, Mar 09 2012

The offset is chosen as "0" to match the generalized or compositional Bernoulli numbers.
Following [Blandin and Diaz], we can generalize a subset of Bernoulli numbers to comply with the origin of the triangle (the Pascal matrix A007318 beheaded once: (A074909), twice: (this triangle), and so on...); and a corresponding Bernoulli sequence that equals the inverse of the triangle, extracting the left border. This procedure done with A074909 results in The Bernoulli numbers (A027641/A026642) starting (1, -1/2, 1/6,...). Done with this triangle we obtain A006568/A006569: (1, -1/3, 1/18, 1/90,...).
A generalized algebraic property of the subset of such triangles and compositional Bernoulli numbers is that the triangle M * [corresponding Bernoulli sequence considered as a vector, V] = [1, 0, 0, 0,...].
The infinite set of generalized Bernoulli number sequences thus generated from variants of Pascal's triangle begins: [(1, -1/2, 1/6,...); (1, -1/3, 1/18,...); (1, -1/4, 1/40,...); (1, -1/5, 1/75,...); where the third term denominators = A002411 (1, 6, 18, 40, 75,...) after the "1".
Row sums of the triangle = A000295 starting (1, 4, 11, 26, 57,...).

			First few rows of the triangle =
1;
1, 3;
1, 4, 6;
1, 5, 10, 10;
1, 6, 15, 20, 15;
1, 7, 21, 35, 35, 21;
1, 8, 28, 56, 70, 56, 28;
1, 9, 36, 84, 126, 126, 84, 36;
1, 10, 45, 120, 210, 252, 210, 120, 45;
1, 11, 55, 165, 330, 462, 462, 30, 165, 55;
...

Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 2nd table is identical to A006569/A006568.

Cf. A000295, A104721, A074909, A027641, A027642, A006568, A006569, A007318, A002411.

Table[Binomial[n+2, k+2], {n, 0, 9}, {k, n , 0, -1}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Doubly beheaded variant of Pascal's triangle in which two rightmost diagonals are deleted.

T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-3*T(n-2,k-2)+T(n-3,k-2)+T(n-3,k-3), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014

A383870 Number of compositions of n such that none of the smallest parts are adjacent.

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 15, 29, 53, 98, 180, 336, 618, 1142, 2110, 3899, 7197, 13283, 24509, 45218, 83396, 153769, 283463, 522449, 962732, 1773742, 3267417, 6018030, 11082693, 20407174, 37572633, 69169726, 127326924, 234362474, 431343281, 793831500, 1460854117

Offset: 0

John Tyler Rascoe, May 13 2025

			a(5) = 9 counts: (1,2,2), (1,3,1), (1,4), (2,1,2), (2,2,1), (2,3), (3,2), (4,1), (5).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A003242, A007318, A011782, A074909, A105039, A238342.

b:= proc(n, i, p) option remember; `if`(i<1, 0,
     `if`(irem(n, i, 'r')=0, p!*binomial(p+1, r), 0)+
      add(b(n-i*j, min(n-i*j, i-1), p+j)/j!, j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..36);  # Alois P. Heinz, May 13 2025

A_x(N) ={Vec(1+sum(j=0,N, sum(i=j+1,N-j, (binomial(i,i-j-1) * x^(j+1) * (x^2/(1-x))^(i-1) )/(1-x^(i+j))))+O('x^N))}
A_x(50)

G.f.: 1 + Sum_{j>=0} Sum_{i>j} (binomial(i,i-j-1) * x^(j+1) * (x^2/(1 - x))^(i-1))/(1 - x^(i+j)).

A227985 Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.

Original entry on oeis.org

1, 0, -1, 0, -1, 1, 0, -1, 1, -1, 0, -1, 1, -1, 0, 0, -1, 1, -5, 0, 1, 0, -1, 1, -1, 0, 1, 0, 0, -1, 1, -7, 0, 7, 0, -1, 0, -1, 1, -2, 0, 7, 0, -2, 0, 0, -1, 1, -3, 0, 7, 0, -1, 0, 3, 0, -1, 1, -5, 0, 1, 0, -1, 0, 1, 0, 0, -1, 1, -11, 0, 11, 0, -11, 0, 11, 0, -5, 0, -1, 1, -1, 0, 11, 0, -22, 0, 33, 0, -5, 0

Offset: 0

Paul Curtz, Aug 02 2013

The n-th row's sum equals the n-th Bernoulli number (with B_1 = -1/2).
Starting from B_0 = 1, the successive B n comes from the equations written with the triangle A074909
1*B_0 +2*B_1 = 0                   -->  B_1 = 0 -1/2
1*B_0 +3*B_1 +3*B_2 = 0            -->  B_2 = 0 -1/3 +1/2
1*B_0 +4*B_1 +6*B_2 +4*B_3 = 0     -->  B_3 = 0 -1/4 +1/2 -1/4,
from the terms at the left-hand side. See A159688.
Main diagonal: 1, -1/2, 1/2, -1/4, 0, 1/12, 0, -1/12, 0, 3/20, 0, -5/12, 0, 691/420,... . After the initial 1, the numerators are given by -A050925.

			Triangle begins:
1,
0, -1,
0, -1, 1,
0, -1, 1, -1,
0, -1, 1, -1, 0,
0, -1, 1, -5, 0, 1,
0  -1, 1, -1, 0, 1,  0,
0, -1, 1, -7, 0, 7,  0, -1,
0, -1, 1, -2, 0, 7,  0, -2, 0, etc.

Cf. A027641/A027642, A074909.

[1] cat [Numerator(-Binomial(n,k)*Bernoulli(k)/n): k in [-1..n-2], n in [2..15]]; // Bruno Berselli, Sep 09 2013

b[0] = 1; b[1] = -1/2; row[0] = {1}; row[1] = {0, -1/2}; row[n_] := Join[{0}, List @@ (-Sum[Binomial[n+1, k]*B[k], {k, 0, n-1}]/(n+1) // Expand) /. B -> b]; b[n_] := Total[row[n]]; Table[row[n] // Numerator, {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 12 2013 *)

t(n, k) = if (n==1, 1, if (k== -1, 0, -bernfrac(k)*binomial(n, k)/n));
tabl(nn) = {for (n = 1, nn, for (k = -1, n-2, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Sep 07 2013

More terms from Jean-François Alcover, Aug 12 2013

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 *)

cp's OEIS Frontend

A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

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A193465 Row sums of triangle A061312.

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A071761 Dealing cards in a game of solitaire.

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A168557 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial (-1)^n*((x + 1)^n - x^n + 1), 0 <= k <= max(0, n - 1).

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A199011 Triangle T(n,k), read by rows, given by (1,1,-1,1,0,0,0,0,0,0,0,...) DELTA (0,1,0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A209518 Triangle by rows, reversal of A104712.

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A383870 Number of compositions of n such that none of the smallest parts are adjacent.

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A227985 Numerators of the fractional triangle T(n,k) = binomial(n-1,k)*B_k for 0 <= k < n.

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