A034264 -id:A034264 - cp's OEIS frontend

A034261 Infinite square array f(a,b) = C(a+b,b+1)(ab+a+1)/(b+2), a, b >= 0, read by antidiagonals. Equivalently, triangular array T(n,k) = f(k,n-k), 0 <= k <= n, read by rows.

0, 0, 1, 0, 1, 3, 0, 1, 5, 6, 0, 1, 7, 14, 10, 0, 1, 9, 25, 30, 15, 0, 1, 11, 39, 65, 55, 21, 0, 1, 13, 56, 119, 140, 91, 28, 0, 1, 15, 76, 196, 294, 266, 140, 36, 0, 1, 17, 99, 300, 546, 630, 462, 204, 45, 0, 1, 19, 125, 435, 930, 1302, 1218, 750, 285, 55

Offset: 0

Clark Kimberling

f(h,k) = number of paths consisting of steps from (0,0) to (h,k) using h unit steps right, k+1 unit steps up and 1 unit step down, in some order, with first step not down and no repeated points.

			Triangle begins:
  0;
  0, 1;
  0, 1, 3;
  0, 1, 5,  6;
  0, 1, 7, 14, 10;
  ...
As a square array,
  [ 0  0  0   0   0 ...]
  [ 1  1  1   1   1 ...]
  [ 3  5  7   9  11 ...]
  [ 6 14 25  39  56 ...]
  [10 30 65 119 196 ...]
  [...      ...     ...]

Cf. A001787 (row sums), A000330(n) = f(n,1).

Cf. A034263, A034264, A034265, A034267 - A034275 for diagonals n -> f(n,n+k), for several fixed k.

A034261 := proc(n, k) binomial(n, n-k+1)*(k+(k-1)/(k-n-2)); end proc; # argument indices of the triangle

Flatten[Table[Binomial[n,n-k+1](k+(k-1)/(k-n-2)),{n,0,15},{k,0,n}]] (* Harvey P. Dale, Jan 11 2013 *)

f(h,k)=binomial(h+k,k+1)*(k*h+h+1)/(k+2)

tabl(nn) = for (n=0, nn, for (k=0, n, print1(binomial(n, n-k+1)*(k+(k-1)/(k-n-2)), ", ")); print()); \\ Michel Marcus, Mar 20 2015

Another formula: f(h,k) = binomial(h+k,k+1) + Sum{C(i+j-1, j)*C(h+k-i-j, k-j+1): i=1, 2, ..., h-1, j=1, 2, ..., k+1}

Entry revised by N. J. A. Sloane, Apr 21 2000. The formula for f in the definition was found by Michael Somos.

Edited by M. F. Hasler, Nov 08 2017

A107600 Column 5 of array illustrated in A089574 and related to A034261.

Original entry on oeis.org

1, 18, 101, 357, 978, 2274, 4711, 8954, 15915, 26806, 43197, 67079, 100932, 147798, 211359, 296020, 406997, 550410, 733381, 964137, 1252118, 1608090, 2044263, 2574414, 3214015, 3980366, 4892733, 5972491, 7243272, 8731118, 10464639

Offset: 9

Alford Arnold, May 17 2005

The sequences in A089574 count ordered partitions. Sequence A001296 can be associated with 9 = 3+3+3. Six times sequence A005585, associated with 10 = 3+3+2+2. The other three sequences comprising A107600 are generated in A034261 and can be associated with 10 = 5 + 5 = 4 + 4 + 2 = 2 + 2 + 2 + 2 + 2.

			A107600(n) can be constructed from five other sequences as follows:
1...7...25...65...140.......A001296
....1...11...56...196.......A034264
....6...42..162...462.......6.*.A005585.
....3...18...60...150.......A006011
....1....5...14....30.......A000330
therefore
1..18..101..357...978.......A107600

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A034261, A089574, A000330, A001296, A005585, A006011, A034264.

a:= n-> `if` (n<9, 0, (92292 +(-6580 +(-5745 +(1535 +(-147+5*n) *n) *n) *n) *n) *n /720 -218): seq(a(n), n=9..45); # Alois P. Heinz, Nov 06 2009

Select[CoefficientList[Series[(x^5-5x^4+7x^3+4x^2-11x-1)x^9/(x-1)^7, {x,0,50}],x],#>0&] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1,18,101,357,978,2274,4711},42] (* Harvey P. Dale, May 01 2011 *)

G.f.: (x^5 -5*x^4 +7*x^3 +4*x^2 -11*x -1) *x^9 /(x-1)^7. - Alois P. Heinz, Nov 06 2009

More terms from Alois P. Heinz, Nov 06 2009

A051946 Expansion of g.f.: (1+4*x)/(1-x)^7.

Original entry on oeis.org

1, 11, 56, 196, 546, 1302, 2772, 5412, 9867, 17017, 28028, 44408, 68068, 101388, 147288, 209304, 291669, 399399, 538384, 715484, 938630, 1216930, 1560780, 1981980, 2493855, 3111381, 3851316, 4732336, 5775176, 7002776, 8440432

Offset: 0

Barry E. Williams, Dec 20 1999

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 18 2005

Equals row sums of triangle A143130, and binomial transform of {1, 10, 35, 60, 55, 26, 5, 0, 0, 0, ...}. - Gary W. Adamson, Jul 27 2008

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p.233, # 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Partial sums of A027800.
Cf. A093562 ((5, 1) Pascal, column m=6).
Cf. A143130.
Cf. similar sequences listed in A254142.

List([0..40], n-> (5*n+6)*Binomial(n+5,5)/6); # G. C. Greubel, Aug 28 2019

[(5*n+6)*Binomial(n+5,5)/6: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720: seq(a(n),n=0..35); # Emeric Deutsch

CoefficientList[Series[(1+4x)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

vector(40, n, (5*n+1)*binomial(n+4,5)/6) \\ G. C. Greubel, Aug 28 2019

[(5*n+6)*binomial(n+5,5)/6 for n in (0..40)] # G. C. Greubel, Aug 28 2019

a(n) = binomial(n+5,5)*(5*n+6)/6.
a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(5*n+6)/720. - Emeric Deutsch, Jun 18 2005
a(n) = A034264(n+1). - R. J. Mathar, Oct 14 2008

Corrected and extended by Emeric Deutsch, Jun 18 2005

cp's OEIS Frontend

A034261 Infinite square array f(a,b) = C(a+b,b+1)*(a*b+a+1)/(b+2), a, b >= 0, read by antidiagonals. Equivalently, triangular array T(n,k) = f(k,n-k), 0 <= k <= n, read by rows.

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A107600 Column 5 of array illustrated in A089574 and related to A034261.

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A051946 Expansion of g.f.: (1+4*x)/(1-x)^7.

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A034261 Infinite square array f(a,b) = C(a+b,b+1)(ab+a+1)/(b+2), a, b >= 0, read by antidiagonals. Equivalently, triangular array T(n,k) = f(k,n-k), 0 <= k <= n, read by rows.