A051797 -id:A051797 - cp's OEIS frontend

A051878 Partial sums of A051797.

1, 13, 63, 203, 518, 1134, 2226, 4026, 6831, 11011, 17017, 25389, 36764, 51884, 71604, 96900, 128877, 168777, 217987, 278047, 350658, 437690, 541190, 663390, 806715, 973791, 1167453, 1390753, 1646968, 1939608, 2272424, 2649416, 3074841, 3553221, 4089351, 4688307, 5355454, 6096454

Offset: 0

Barry E. Williams, Dec 14 1999

Convolution of triangular numbers (A000217) and decagonal numbers (A001107). [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 to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000217, A001107, A051797.

Cf. A093565 ((8, 1) Pascal, column m=5).

List([0..40], n-> (8*n+5)*Binomial(n+4,4)/5); # G. C. Greubel, Aug 30 2019

[(8*n+5)*Binomial(n+4,4)/5: n in [0..40]]; // G. C. Greubel, Aug 30 2019

seq((8*n+5)*binomial(n+4,4)/5, n=0..40); # G. C. Greubel, Aug 30 2019

Table[(8*n+5)*Binomial[n+4,4]/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011, modified by G. C. Greubel, Aug 30 2019 *)

vector(40, n, (8*n-3)*binomial(n+3,4)/5) \\ G. C. Greubel, Aug 30 2019

[(8*n+5)*binomial(n+4,4)/5 for n in (0..30)] # G. C. Greubel, Aug 30 2019

a(n) = binomial(n+4, 4)*(8*n+5)/5.
G.f.: (1+7*x)/(1-x)^6.
E.g.f.: (120 +*1440*x +2280*x^2 +1040*x^3 +165*x^4 +8*x^5)*exp(x)/120. - G. C. Greubel, Aug 30 2019

Terms a(28) onward added by G. C. Greubel, Aug 30 2019

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 15, 20, 1, 7, 20, 35, 35, 1, 8, 25, 50, 70, 56, 1, 9, 30, 65, 105, 126, 84, 1, 10, 35, 80, 140, 196, 210, 120, 1, 11, 40, 95, 175, 266, 336, 330, 165, 1, 12, 45, 110, 210, 336, 462, 540, 495, 220, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 286

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array, n >= 0, k >= 0, begins
1 4 10 20  35  56 ...
1 5 15 35  70 126 ...
1 6 20 50 105 196 ...
1 7 25 65 140 266 ...
1 8 30 80 175 336 ...

Index to sequences related to pyramidal numbers

Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.
Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).
See A257200 for another version of the array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5,x,11),x,n),n,0,11),k,-1,10)

VECTOR(VECTOR(comb(k+3,3)+comb(k+3,4)n, k, 0, 11), n, 0, 11)

A080852 := proc(n,k)
        binomial(k+4,4)+(n-1)*binomial(k+3,4) ;
end proc:
seq( seq(A080852(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

T[n_, k_] := Binomial[k+3, 3] + Binomial[k+3, 4]n;
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.
T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.
G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.
T(n,k) = sum_{j=0..k} A080851(n,j). - R. J. Mathar, Jul 28 2016

A093565 (8,1) Pascal triangle.

Original entry on oeis.org

1, 8, 1, 8, 9, 1, 8, 17, 10, 1, 8, 25, 27, 11, 1, 8, 33, 52, 38, 12, 1, 8, 41, 85, 90, 50, 13, 1, 8, 49, 126, 175, 140, 63, 14, 1, 8, 57, 175, 301, 315, 203, 77, 15, 1, 8, 65, 232, 476, 616, 518, 280, 92, 16, 1, 8, 73, 297, 708, 1092, 1134, 798, 372, 108, 17, 1, 8, 81, 370, 1005

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(8;n,m) gives in the columns m>=1 the figurate numbers based on A017077, including the decagonal numbers A001107,(see the W. Lang link).
This is the eighth member, d=8, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-4, for d=1..7.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). 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)=(1+7*z)/(1-(1+x)*z).
The SW-NE diagonals give A022098(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 7. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins
  [1];
  [8,  1];
  [8,  9,  1];
  [8, 17, 10,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Lang, First 10 rows and array of figurate numbers .

Row sums: A005010(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 7 for n=2 and 0 else.
The column sequences give for m=1..9: A017077, A001107 (decagonal), A007585, A051797, A051878, A050404, A052226, A056001, A056122.
Cf. A093644 (d=9).

a093565 n k = a093565_tabl !! n !! k
a093565_row n = a093565_tabl !! n
a093565_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [8, 1]
-- Reinhard Zumkeller, Aug 31 2014

a(n, m)=F(8;n-m, m) for 0<= m <= n, otherwise 0, with F(8;0, 0)=1, F(8;n, 0)=8 if n>=1 and F(8;n, m):=(8*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=8 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+7*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 7*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 17*x + 10*x^2/2! + x^3/3!) = 8 + 25*x + 52*x^2/2! + 90*x^3/3! + 140*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Original entry on oeis.org

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A213835 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 4n-7+4h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 5, 22, 19, 9, 50, 46, 31, 13, 95, 90, 70, 43, 17, 161, 155, 130, 94, 55, 21, 252, 245, 215, 170, 118, 67, 25, 372, 364, 329, 275, 210, 142, 79, 29, 525, 516, 476, 413, 335, 250, 166, 91, 33, 715, 705, 660, 588, 497, 395

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A172078.
Antidiagonal sums: A051797.
Row 1, (1,2,3,4,5,...)**(1,5,9,13,...): A002412.
Row 2, (1,2,3,4,5,...)**(5,9,13,17,...): (4*k^3 + 15*k^2 - 11*k)/6.
Row 3, (1,2,3,4,5,...)**(9,13,17,21,...): (4*k^3 + 27*k^2 - 23*k)/6
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....7....22....50....95
5....19...46....90....155
9....31...70....130...215
13...43...94....170...275
17...55...118...210...335
21...67...142...250...395

Clark Kimberling, Antidiagonals n = 1..60, flattened.

Cf. A212500.

Cf. A304659 (first lower diagonal).

b[n_]:=n;c[n_]:=4n-3;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213835 *)
Table[t[n,n],{n,1,40}] (* A172078 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A051797 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*((4*n-3) + (4*n-7)*x) and g(x) = (1-x)^4.

A264850 a(n) = n(n + 1)(n + 2)(7n - 5)/12.

Original entry on oeis.org

0, 1, 18, 80, 230, 525, 1036, 1848, 3060, 4785, 7150, 10296, 14378, 19565, 26040, 34000, 43656, 55233, 68970, 85120, 103950, 125741, 150788, 179400, 211900, 248625, 289926, 336168, 387730, 445005, 508400, 578336, 655248, 739585, 831810, 932400, 1041846

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of 16-gonal (or hexadecagonal) pyramidal numbers. Therefore, this is the case k=7 of the general formula n*(n + 1)*(n + 2)*(k*n - k + 2)/12, which is related to 2*(k+1)-gonal pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A172076.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12: A000292 (k=0), A002415 (which arises from k=1), A002417 (k=2), A002419 (k=3), A051797 (k=4), A051799 (k=5), A220212 (k=6), this sequence (k=7), A264851 (k=8), A264852 (k=9).

[n*(n+1)*(n+2)*(7*n-5)/12: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (n + 2) (7 n - 5)/12, {n, 0, 50}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,18,80,230},40] (* Harvey P. Dale, Sep 27 2018 *)

a(n)=n*(n+1)*(n+2)*(7*n-5)/12 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 13*x)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A172076(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

cp's OEIS Frontend

A051878 Partial sums of A051797.

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A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

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A080852 Square array of 4D pyramidal numbers, read by antidiagonals.

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Derive

Derive

Maple

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A093565 (8,1) Pascal triangle.

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Haskell

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A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

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Magma

Magma

Mathematica

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A213835 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 4*n-7+4*h, n>=1, h>=1, and ** = convolution.

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A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A213835 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 4n-7+4h, n>=1, h>=1, and ** = convolution.

A264850 a(n) = n(n + 1)(n + 2)(7n - 5)/12.