A034263 -id:A034263 - cp's OEIS frontend

A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

1, 6, 1, 15, 7, 1, 28, 22, 8, 1, 45, 50, 30, 9, 1, 66, 95, 80, 39, 10, 1, 91, 161, 175, 119, 49, 11, 1, 120, 252, 336, 294, 168, 60, 12, 1, 153, 372, 588, 630, 462, 228, 72, 13, 1, 190, 525, 960, 1218, 1092, 690, 300, 85, 14, 1, 231, 715, 1485, 2178, 2310, 1782, 990, 385, 99, 15, 1

Offset: 0

Gary W. Adamson, Nov 24 2006

Left border = A000384, hexagonal numbers. The following columns are A002412, A002417, A034263, A051947, ...
Row sums = (1, 7, 23, 59, 135, 291, ...) = A126284.
A125232 is the analogous triangle for the pentagonal numbers.

			First few rows of the triangle:
   1;
   6,   1;
  15,   7,   1;
  28,  22,   8,   1;
  45,  50,  30,   9,  1;
  66,  95,  80,  39, 10,  1;
  91, 161, 175, 119, 49, 11, 1;
  ...
Example: (5,3) = 80 = 30 + 50 = (4,3) + (4,2).

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1964, p. 189.

Cf. A000384, A002412, A002417, A034263, A051947, A125232.

A000384Psum:= proc(n,k) coeftayl( x*(1+3*x)/(1-x)^(3+k),x=0,n) ; end: A125233 := proc(n,k) A000384Psum(n-k,k) ; end: for n from 1 to 15 do for k from 0 to n -1 do printf("%d,",A125233(n,k)) ; od: od: # R. J. Mathar, May 03 2008

T[n_, k_] := T[n, k] = Which[k == 0, n (2 n - 1), 1 <= k < n, T[n - 1, k] + T[n - 1, k - 1], True, 0];
Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Sep 14 2023, after R. J. Mathar *)

T(n,0)=A000384(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>1. - R. J. Mathar, May 03 2008

Edited and extended by R. J. Mathar, May 03 2008, and M. F. Hasler, Sep 29 2012

A059599 Expansion of (3+x)/(1-x)^6.

Original entry on oeis.org

3, 19, 69, 189, 434, 882, 1638, 2838, 4653, 7293, 11011, 16107, 22932, 31892, 43452, 58140, 76551, 99351, 127281, 161161, 201894, 250470, 307970, 375570, 454545, 546273, 652239, 774039, 913384, 1072104, 1252152, 1455608, 1684683, 1941723

Offset: 0

Wolfdieter Lang, Feb 02 2001

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

Cf. A034263.

f[n_]:=Binomial[n+4,4]*(15+4*n)/5; Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

a(n) = binomial(n+4, 4)*(15+4*n)/5.
G.f.: (3+x)/(1-x)^6.
a(-n-4) = -A034263(n). - Bruno Berselli, Aug 23 2011
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Apr 16 2023

More terms from Vladimir Joseph Stephan Orlovsky, May 30 2010

A034273 a(n) = binomial(2n+6,n+7)(n^2+7*n+1)/(n+8) = f(n,n+6) where f is given in A034261.

Original entry on oeis.org

0, 1, 19, 186, 1365, 8540, 48348, 255816, 1289739, 6273135, 29683225, 137447310, 625482585, 2806282440, 12443418600, 54633668400, 237871030860, 1028260382994, 4417404967206, 18874729444340, 80265980340370, 339907420551336, 1434074601137640, 6030288337651760

Offset: 0

Clark Kimberling

Michael De Vlieger, Table of n, a(n) for n = 0..1655

Cf. A034261 (main entry), A034263 - A034275 (other columns and diagonals n -> f(n,n+k)).

f[a_, b_] := Binomial[a + b, b + 1] (a b + a + 1)/(b + 2); Array[f[#, # + 6] &, 22, 0] (* Michael De Vlieger, Nov 08 2017 *)

A034273(n)=binomial(2*n+6, n+7)*(n^2+7*n+1)/(n+8) \\ M. F. Hasler, Nov 08 2017

a(n) ~ 2^(2*n+6) * sqrt(n/Pi). - Amiram Eldar, Sep 04 2025

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

Edited by M. F. Hasler, Nov 08 2017

A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

Original entry on oeis.org

1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Triangular Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

/* From definition: */ P:=func; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

[(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

O.g.f.: (1 + 11*x)/(1 - x)^6.
E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.
Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

Edited by Bruno Berselli, Apr 18 2016

A143129 Triangle read by rows, T(n,k) = sum {j=k..n} A000292(j) = A000012 * (A000292 * 0^(n-k)) * A000012, 1<=k<=n.

Original entry on oeis.org

1, 5, 4, 15, 14, 10, 35, 34, 30, 20, 70, 69, 65, 55, 35, 126, 125, 121, 111, 91, 56, 210, 209, 205, 195, 175, 140, 84, 330, 329, 325, 315, 295, 260, 204, 120, 495, 494, 490, 480, 460, 425, 369, 285, 165, 715, 714, 710, 700, 680, 645, 589, 505, 385, 220, 1001, 1000

Offset: 1

Gary W. Adamson, Jul 26 2008

Row sums = A034263: (1, 9, 39, 119, 294,...).

			First few rows of the triangle =
1;
5, 4;
15, 14, 10;
35, 34, 30, 20;
70, 69, 65, 55, 35;
126, 125, 121, 111, 91, 56;
210, 209l 205, 195, 175, 140, 84;
...
T(5,3) = 65 = (10 + 20 + 35), where A000292 = (1, 4, 10, 20, 35,...)

Cf. A000292, A034263.

Triangle read by rows, T(n,k) = sum {j=k..n} A000292(j) = A000012 * (A000292 * 0^(n-k)) * A000012, 1<=k<=n

cp's OEIS Frontend

A125233 Triangle T(n,k) read by rows, the (n-k)-th term of the k times repeated partial sum of the hexagonal numbers, 0 <= k < n, 0 < n.

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A059599 Expansion of (3+x)/(1-x)^6.

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A034273 a(n) = binomial(2*n+6,n+7)*(n^2+7*n+1)/(n+8) = f(n,n+6) where f is given in A034261.

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A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

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Magma

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A143129 Triangle read by rows, T(n,k) = sum {j=k..n} A000292(j) = A000012 * (A000292 * 0^(n-k)) * A000012, 1<=k<=n.

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A034273 a(n) = binomial(2n+6,n+7)(n^2+7*n+1)/(n+8) = f(n,n+6) where f is given in A034261.