A093563 -id:A093563 - cp's OEIS frontend

A096956 Pascal (1,6) triangle.

6, 1, 6, 1, 7, 6, 1, 8, 13, 6, 1, 9, 21, 19, 6, 1, 10, 30, 40, 25, 6, 1, 11, 40, 70, 65, 31, 6, 1, 12, 51, 110, 135, 96, 37, 6, 1, 13, 63, 161, 245, 231, 133, 43, 6, 1, 14, 76, 224, 406, 476, 364, 176, 49, 6, 1, 15, 90, 300, 630, 882, 840, 540, 225, 55, 6, 1, 16, 105, 390, 930

Offset: 0

Wolfdieter Lang, Aug 13 2004

Except for the first row this is the row reversed (6,1)-Pascal triangle A093563.
This is the sixth member, q=6, in the family of (1,q) Pascal triangles: A007318 (Pascal (q=1)), A029635 (q=2) (but with a(0,0)=2, not 1), A095660 (q=3), A095666 (q=4), A096940 (q=5).
This is an example of a Riordan triangle (see A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group) with o.g.f. of column no. m of the type g(x)*(x*f(x))^m with f(0)=1. Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=g(z)/(1-x*z*f(z)). Here: g(x)=(6-5*x)/(1-x), f(x)=1/(1-x), hence G(z,x)=(6-5*z)/(1-(1+x)*z).
The SW-NE diagonals give Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k) = A022097(n-2), n >= 2, with n=1 value 6. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins:
  [0]  6;
  [1]  1,  6;
  [2]  1,  7,  6;
  [3]  1,  8, 13,  6;
  [4]  1,  9, 21, 19,  6;
  [5]  1, 10, 30, 40, 25,  6;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Wolfdieter Lang, First 10 rows.

Row sums: A005009(n-1), n>=1, 6 if n=0; g.f.: (6-5*x)/(1-2*x). Alternating row sums are [6, -5, followed by 0's].

Column sequences (without leading zeros) give for m=1..9, with n >= 0: A000027(n+6), A056115, A096957-9, A097297-A097300.

a(n,k):=piecewise(n=0,6,0Mircea Merca, Apr 08 2012

A096956[n_, k_] := If[n == k, 6, (5*k/n + 1)*Binomial[n, k]];
Table[A096956[n, k], {n, 0, 12}, {k, 0, n}] (* Paolo Xausa, Apr 14 2025 *)

Recursion: a(n,m)=0 if m > n, a(0,0) = 6; a(n,0) = 1 if n >= 1; a(n,m) = a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (6-5*x)/(1-x)^(m+1), m >= 0.
a(n,k) = (1+5*k/n)*binomial(n,k), for n > 0. - Mircea Merca, Apr 08 2012

A051843 Partial sums of A002419.

Original entry on oeis.org

0, 1, 11, 51, 161, 406, 882, 1722, 3102, 5247, 8437, 13013, 19383, 28028, 39508, 54468, 73644, 97869, 128079, 165319, 210749, 265650, 331430, 409630, 501930, 610155, 736281, 882441, 1050931, 1244216, 1464936, 1715912, 2000152, 2320857, 2681427

Offset: 0

Barry E. Williams, Dec 13 1999

5-dimensional form of octagonal-based pyramidal numbers. - Derek I. Thomas (dithom02(AT)louisville.edu), Jun 30 2007
Convolution of triangular numbers (A000217) and octagonal numbers (A000567). [Bruno Berselli, Jul 21 2015]
Also the number of 4-cycles in the (n+2)-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
H. J. Ryser, Combinatorial Mathematics, Carus Mathematical Monographs No. 14, John Wiley and Sons, 1963, pp. 1-8.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Octagonal Number
Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002419; A000217, A000567.
Cf. A093563 ((6, 1) Pascal, column m=5).
Cf. A034827 (3-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

Join[{0}, Accumulate[LinearRecurrence[{5, -10, 10, -5, 1},{1, 10, 40, 110, 245}, 40]]] (* Harvey P. Dale, Nov 30 2014 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 11, 51, 161, 406}, 40] (* Harvey P. Dale, Nov 30 2014 *)
Table[(6 n - 1) Binomial[n + 3, 4]/5, {n, 0, 20}] (* Eric W. Weisstein, Aug 10 2017 *)

a(n) = C(n+3,4) * (6*n-1)/5
G.f.: x*(1+5*x)/(1-x)^6.
a(n) = n*(n+1)*(n+2)*(n+3)*(6n-1)/120. - Derek I. Thomas (dithom02(AT)louisville.edu), Jun 30 2007

a(1) corrected by Gael Linder (linder.gael(AT)wanadoo.fr), Oct 31 2007

a(0) prepended by Joerg Arndt, Jun 26 2013

A034265 a(n) = binomial(n+6,6)(6n+7)/7.

Original entry on oeis.org

1, 13, 76, 300, 930, 2442, 5676, 12012, 23595, 43615, 76648, 129064, 209508, 329460, 503880, 751944, 1097877, 1571889, 2211220, 3061300, 4177030, 5624190, 7480980, 9839700, 12808575, 16513731, 21101328, 26739856, 33622600, 41970280

Offset: 0

Clark Kimberling

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

a(n)=f(n, 5) where f is given in A034261.
Partial sums of A027810.
Cf. A093563 ((6, 1) Pascal, column m=7).
Cf. similar sequences listed in A254142.

List([0..30], n-> (6*n+7)*Binomial(n+6,6)/7); # G. C. Greubel, Aug 28 2019

[(6*n+7)*Binomial(n+6,6)/7: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

seq((6*n+7)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Aug 28 2019

Accumulate[Table[(n+1)Binomial[n+5,5],{n,0,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {1,13,76,300,930,2442,5676, 12012}, 30] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1+5x)/(1-x)^8, {x,0,40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

a(n)=(6*n/7+1)*binomial(n+6,6) \\ Charles R Greathouse IV, Oct 07 2015

[(6*n+7)*binomial(n+6,6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019

G.f.: (1+5*x)/(1-x)^8.

a(0)=1, a(1)=13, a(2)=76, a(3)=300, a(4)=930, a(5)=2442, a(6)=5676, a(7)=12012, a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). - Harvey P. Dale, Jul 29 2014

Corrected and extended by N. J. A. Sloane, Apr 21 2000

A054487 a(n) = (3n+4)binomial(n+7, 7)/4.

Original entry on oeis.org

1, 14, 90, 390, 1320, 3762, 9438, 21450, 45045, 88660, 165308, 294372, 503880, 833340, 1337220, 2089164, 3187041, 4758930, 6970150, 10031450, 14208480, 19832670, 27313650, 37153350, 49961925, 66475656, 87576984, 114316840

Offset: 0

Barry E. Williams, May 06 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A034265.

Cf. A093563 ((6, 1) Pascal, column m=8).

List([0..40], n-> (3*n+4)*Binomial(n+7, 7)/4 ); # G. C. Greubel, Jan 19 2020

[((3*n+4)*Binomial(n+7,7))/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

seq( (3*n+4)*binomial(n+7,7)/4, n=0..40); # G. C. Greubel, Jan 19 2020

CoefficientList[Series[(1+5x)/(1-x)^9, {x,0,40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[6*Binomial[n+8,8] -5*Binomial[n+7,7], {n,0,40}] (* G. C. Greubel, Jan 19 2020 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,14,90,390,1320,3762,9438,21450,45045},30] (* Harvey P. Dale, Jul 19 2022 *)

a(n) = (3*n+4)*binomial(n+7, 7)/4; \\ Michel Marcus, Sep 07 2017

[(3*n+4)*binomial(n+7, 7)/4 for n in (0..40)] # G. C. Greubel, Jan 19 2020

G.f.: (1+5*x)/(1-x)^9.
From G. C. Greubel, Jan 19 2020: (Start)
a(n) = 6*binomial(n+8, 8) - 5*binomial(n+7, 7).
E.g.f.: (20160 +262080*x +635040*x^2 +540960*x^3 +205800*x^4 +38808*x^5 +3724*x^6 +172*x^7 +3*x^8)*exp(x)/20160. (End)
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Wesley Ivan Hurt, Jun 07 2021

Corrected and extended by James Sellers, May 10 2000

A055848 Expansion of (1+5*x)/(1-x)^10.

Original entry on oeis.org

1, 15, 105, 495, 1815, 5577, 15015, 36465, 81510, 170170, 335478, 629850, 1133730, 1967070, 3304290, 5393454, 8580495, 13339425, 20309575, 30341025, 44549505, 64382175, 91695825, 128849175, 178811100, 245286756, 332863740, 447180580

Offset: 0

Barry E. Williams, Jun 03 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A054487.

Cf. A093563 ((6, 1) Pascal, column m=9). Partial sums of A054487.

Table[(2n+3)Binomial[n+8,8]/3,{n,0,30}] (* Harvey P. Dale, Aug 20 2011 *)

Vec((1+5*x)/(1-x)^10 + O(x^100)) \\ Altug Alkan, Mar 13 2016

a(n)=(2n+3)*C(n+8, 8)/3. G.f.(x)=(1+5x)/(1-x)^10.

cp's OEIS Frontend

A096956 Pascal (1,6) triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A051843 Partial sums of A002419.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A034265 a(n) = binomial(n+6,6)*(6*n+7)/7.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A054487 a(n) = (3*n+4)*binomial(n+7, 7)/4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A055848 Expansion of (1+5*x)/(1-x)^10.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A034265 a(n) = binomial(n+6,6)(6n+7)/7.

A054487 a(n) = (3n+4)binomial(n+7, 7)/4.