A093644 -id:A093644 - cp's OEIS frontend

A051798 a(n) = (n+1)(n+2)(n+3)*(9n+4)/24.

1, 13, 55, 155, 350, 686, 1218, 2010, 3135, 4675, 6721, 9373, 12740, 16940, 22100, 28356, 35853, 44745, 55195, 67375, 81466, 97658, 116150, 137150, 160875, 187551, 217413, 250705, 287680, 328600, 373736, 423368, 477785, 537285

Offset: 0

Barry E. Williams, Dec 11 1999

Partial sums of A007586.

Convolution of A000027 with A051682 (excluding 0). - Bruno Berselli, Dec 07 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A007586, A051682.
Cf. A093644 ((9, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

/* A000027 convolved with A051682 (excluding 0): */ A051682:=func; [&+[(n-i+1)*A051682(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,13,55,155,350},40] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=(n+1)*(n+2)*(n+3)*(9*n+4)/24 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = C(n+3, 3)*(9*n+4)/4.
G.f.: (1+8*x)/(1-x)^5.
a(0)=1, a(1)=13, a(2)=55, a(3)=155, a(4)=350, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Aug 19 2012
a(n) = A080852(9,n). - R. J. Mathar, Jul 28 2016

A056003 a(n) = (n+1)*binomial(n+8, 8).

Original entry on oeis.org

1, 18, 135, 660, 2475, 7722, 21021, 51480, 115830, 243100, 481338, 906984, 1637610, 2848860, 4796550, 7845024, 12503007, 19468350, 29683225, 44401500, 65270205, 94427190, 134617275, 189329400, 262957500, 360988056, 490217508, 659002960, 877549860, 1158240600

Offset: 0

Barry E. Williams, Jun 12 2000

Original name: A second-order recursive sequence.

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Partial sums of A056117.
Cf. A093644 ((9, 1) Pascal, column m=9).
Cf. A000142, A007318, A052206, A245334, A254142 (partial sums).

a056003 n = (n + 1) * a007318' (n + 8) 8
-- Reinhard Zumkeller, Aug 31 2014

a:=n->(sum((numbcomp(n,9)), j=9..n)):seq(a(n), n=9..35); # Zerinvary Lajos, Aug 26 2008

a[n_] := (n+1)*Binomial[n+8, 8]; Array[a, 50, 0] (* Amiram Eldar, Jan 15 2023 *)

a(n) = (n+1)*binomial(n+8, 8) \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+8*x)/(1-x)^10.
a(n) = A245334(n+8,8)/A000142(8). - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi^2/3 - 266681/22050.
Sum_{n>=0} (-1)^n/a(n) = 2*Pi^2/3 - 38656*log(2)/105 + 611409/2450. (End)

A050405 Partial sums of A051879.

Original entry on oeis.org

1, 15, 84, 308, 882, 2142, 4620, 9108, 16731, 29029, 48048, 76440, 117572, 175644, 255816, 364344, 508725, 697851, 942172, 1253868, 1647030, 2137850, 2744820, 3488940, 4393935, 5486481, 6796440, 8357104

Offset: 0

Barry E. Williams, Dec 21 1999

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

G. C. Greubel, 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).

Cf. A051879.

Cf. A093644 ((9, 1) Pascal, column m=6).

List([0..40], n-> Binomial(n+5, 5)*(3*n+2)/2); # G. C. Greubel, Oct 30 2019

[Binomial(n+5, 5)*(3*n+2)/2: n in [0..40]]; // G. C. Greubel, Oct 30 2019

seq(binomial(n+5, 5)*(3*n+2)/2, n=0..40); # G. C. Greubel, Oct 30 2019

Accumulate[Accumulate[Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}]]] (* Harvey P. Dale, Aug 19 2012 *)

vector(41, n, binomial(n+4, 5)*(3*n-1)/2) \\ G. C. Greubel, Oct 30 2019

[binomial(n+5, 5)*(3*n+2)/2 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = binomial(n+5, 5)*(3*n + 2)/2.
G.f.: (1+8*x)/(1-x)^7.
E.g.f.: (240 +3360*x +6600*x^2 +4000*x^3 +950*x^4 +92*x^5 +3* x^6) *exp(x)/240. - G. C. Greubel, Oct 30 2019

Corrected by T. D. Noe, Nov 09 2006

A051879 Partial sums of A051798.

Original entry on oeis.org

1, 14, 69, 224, 574, 1260, 2478, 4488, 7623, 12298, 19019, 28392, 41132, 58072, 80172, 108528, 144381, 189126, 244321, 311696, 393162, 490820, 606970, 744120, 904995, 1092546, 1309959, 1560664

Offset: 0

Barry E. Williams, Dec 14 1999

Convolution of triangular numbers (A000217) and 11-gonal numbers (A051682). [Bruno Berselli, Jul 21 2015]

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A051798; A000217, A051682.
Cf. A093644((9, 1) Pascal, column m=5).
Cf. A050405.

Accumulate[Table[(n+1)(n+2)(n+3)(9n+4)/24,{n,0,40}]] (* Harvey P. Dale, Aug 19 2012 *)

a(n) = C(n+4, 4)*(9n+5)/5.

G.f.: (1+8*x)/(1-x)^6.

A052206 Partial sums of A050405.

Original entry on oeis.org

1, 16, 100, 408, 1290, 3432, 8052, 17160, 33891, 62920, 110968, 187408, 304980, 480624, 736440, 1100784, 1609509, 2307360, 3249532, 4503400, 6150430, 8288280, 11033100, 14522040, 18915975

Offset: 0

Barry E. Williams, Jan 28 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A050405.

Cf. A093644 ((9, 1) Pascal, column m=7).

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,16,100,408,1290,3432,8052,17160},30] (* Harvey P. Dale, May 28 2018 *)

a(n) = (9n+7)*C(n+6, 6)/7.

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

A056117 Expansion of (1+8*x)/(1-x)^9.

Original entry on oeis.org

1, 17, 117, 525, 1815, 5247, 13299, 30459, 64350, 127270, 238238, 425646, 730626, 1211250, 1947690, 3048474, 4657983, 6965343, 10214875, 14718275, 20868705, 29156985, 40190085, 54712125, 73628100, 98030556, 129229452, 168785452

Offset: 0

Barry E. Williams, Jul 04 2000

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A093644 ((9, 1) Pascal, column m=8). Partial sums of A052206.

List([0..30], n-> (9*n+8)*Binomial(n+7, 7)/8 ); # G. C. Greubel, Jan 18 2020

[(9*n+8)*Binomial(n+7, 7)/8: n in [0..30]]; // G. C. Greubel, Jan 18 2020

seq( (9*n+8)*binomial(n+7, 7)/8, n=0..30); # G. C. Greubel, Jan 18 2020

Table[9*Binomial[n+8,8] -8*Binomial[n+7,7], {n,0,30}] (* G. C. Greubel, Jan 18 2020 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,117,525,1815,5247,13299,30459,64350},30] (* Harvey P. Dale, Nov 23 2022 *)

vector(31, n, (9*n-1)*binomial(n+6, 7)/8) \\ G. C. Greubel, Jan 18 2020

[(9*n+8)*binomial(n+7, 7)/8 for n in (0..30)] # G. C. Greubel, Jan 18 2020

a(n) = (9*n+8)*binomial(n+7, 7)/8.
G.f.: (1+8*x)/(1-x)^9.
From G. C. Greubel, Jan 18 2020: (Start)
a(n) = 9*binomial(n+8,8) - 8*binomial(n+7,7).
E.g.f.: (40320 + 645120*x + 1693440*x^2 + 1505280*x^3 + 588000*x^4 + 112896*x^5 + 10976*x^6 + 512*x^7 + 9*x^8)*exp(x)/40320. (End)

A172185 (9,11) Pascal triangle.

Original entry on oeis.org

1, 9, 11, 9, 20, 11, 9, 29, 31, 11, 9, 38, 60, 42, 11, 9, 47, 98, 102, 53, 11, 9, 56, 145, 200, 155, 64, 11, 9, 65, 201, 345, 355, 219, 75, 11, 9, 74, 266, 546, 700, 574, 294, 86, 11, 9, 83, 340, 812, 1246, 1274, 868, 380, 97, 11, 9, 92, 423, 1152, 2058, 2520, 2142, 1248, 477, 108, 11

Offset: 0

Mark Dols, Jan 28 2010

Sums of NW-SE diagonals give A022114 (apart from first two terms).
Triangle T(n,k), read by rows, given by (9,-8,0,0,0,0,0,0,0,...) DELTA (11,-10,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 09 2011
Row n: Expansion of (9+11x)*(1+x)^(n-1), n > 0. - Philippe Deléham, Oct 09 2011

			Triangle begins:
  1;
  9, 11;
  9, 20,  11;
  9, 29,  31,   11;
  9, 38,  60,   42,   11;
  9, 47,  98,  102,   53,   11;
  9, 56, 145,  200,  155,   64,   11;
  9, 65, 201,  345,  355,  219,   75,   11;
  9, 74, 266,  546,  700,  574,  294,   86,  11;
  9, 83, 340,  812, 1246, 1274,  868,  380,  97,  11;
  9, 92, 423, 1152, 2058, 2520, 2142, 1248, 477, 108, 11;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A022114, A093644, A172179.

T[n_, k_]:= If[n==0, 1, (9 + 2*k/n)*Binomial[n, k]]
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 28 2022 *)

def A172185(n,k): return 9*binomial(n,k) +2*binomial(n-1,k-1) -8*bool(n==0)
flatten([[A172185(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2022

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=9, T(1,1)=11. - Philippe Deléham, Oct 09 2011
G.f.: (1+8*x+10*y*x)/(1-x-y*x). - Philippe Deléham, Apr 13 2012
From G. C. Greubel, Apr 28 2022: (Start)
T(n, k) = 9*binomial(n, k) + 2*binomial(n-1, k-1) with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = 10*2^n - 9*[n=0]. (End)

Corrected and extended by Philippe Deléham, Oct 09 2011

A172283 (-9,11) Pascal triangle.

Original entry on oeis.org

1, -9, 11, -9, 2, 11, -9, -7, 13, 11, -9, -16, 6, 24, 11, -9, -25, -10, 30, 35, 11, -9, -34, -35, 20, 65, 46, 11, -9, -43, -69, -15, 85, 111, 57, 11, -9, -52, -112, -84, 70, 196, 168, 68, 11, -9, -61, -164, -196, -14, 266, 364, 236, 79, 11

Offset: 0

Mark Dols, Jan 30 2010

Triangle T(n,k), read by rows, given by [-9,10,0,0,0,0,0,0,0,...] DELTA [11,-10,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 01 2010

			Triangle begins:
               1
           -9,   11
        -9,    2,   11
     -9,   -7,   13,   11
  -9,  -16,    6,   24,   11

Cf. A093644, A172179, A022114, A000984.

With offset 0: Sum_{k=0..n} T(n,k) = 2^n. - Philippe Deléham, Feb 01 2010
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-9, T(1,1)=1. - Philippe Deléham, Oct 08 2011
G.f.: (1-10*x+10*y*x)/(1-x-y*x). - Philippe Deléham, Apr 13 2012

More terms from Philippe Deléham, Oct 08 2011

cp's OEIS Frontend

A051798 a(n) = (n+1)*(n+2)*(n+3)*(9n+4)/24.

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A056003 a(n) = (n+1)*binomial(n+8, 8).

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A050405 Partial sums of A051879.

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A051879 Partial sums of A051798.

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A052206 Partial sums of A050405.

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A056117 Expansion of (1+8*x)/(1-x)^9.

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A051798 a(n) = (n+1)(n+2)(n+3)*(9n+4)/24.