A051946 -id:A051946 - cp's OEIS frontend

A050484 Partial sums of A051946.

1, 12, 68, 264, 810, 2112, 4884, 10296, 20163, 37180, 65208, 109616, 177684, 279072, 426360, 635664, 927333, 1326732, 1865116, 2580600, 3519230, 4736160, 6296940, 8278920, 10772775, 13884156, 17735472, 22467808, 28242984, 35245760

Offset: 0

Barry E. Williams, Dec 26 1999

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).

Cf. A051946, A093562 ((5, 1) Pascal, column m=7).

[((5*n+7)*Binomial(n+6,6))/7: n in [0..60]]; // Vincenzo Librandi, Jul 30 2014

CoefficientList[Series[(1 + 4 x)/(1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Accumulate[CoefficientList[Series[(1+4x)/(1-x)^7,{x,0,40}],x]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,12,68,264,810,2112,4884,10296},30] (* Harvey P. Dale, Aug 21 2020 *)

a(n)=binomial(n+6, 6)*(5*n+7)/7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = C(n+6, 6)*(5n+7)/7.

G.f.: (1+4*x)/(1-x)^8.

A093562 (5,1) Pascal triangle.

Original entry on oeis.org

1, 5, 1, 5, 6, 1, 5, 11, 7, 1, 5, 16, 18, 8, 1, 5, 21, 34, 26, 9, 1, 5, 26, 55, 60, 35, 10, 1, 5, 31, 81, 115, 95, 45, 11, 1, 5, 36, 112, 196, 210, 140, 56, 12, 1, 5, 41, 148, 308, 406, 350, 196, 68, 13, 1, 5, 46, 189, 456, 714, 756, 546, 264, 81, 14, 1, 5, 51, 235, 645, 1170

Offset: 0

Wolfdieter Lang, Apr 22 2004

This is the fifth member, d=5, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-1, for d=1..4.
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+4*z)/(1-(1+x)*z).
The SW-NE diagonals give A022095(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 4. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
The array F(5;n,m) gives in the columns m >= 1 the figurate numbers based on A016861, including the heptagonal numbers A000566 (see the W. Lang link).
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013
The n-th row polynomial is (4 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Mar 02 2018

			Triangle begins
  [1];
  [5,  1];
  [5,  6,  1];
  [5, 11,  7,  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
P. Bala, A note on the diagonals of a proper Riordan Array
W. Lang, First 10 rows and array of figurate numbers .

Cf. Row sums: A007283(n-1), n>=1, 1 for n=0. A082505(n+1), alternating row sums are 1 for n=0, 4 for n=2 and 0 else.
Column sequences give for m=1..9: A016861, A000566 (heptagonal), A002413, A002418, A027800, A051946, A050484, A052255, A055844.
A007318, A093563 (d=6), A228196, A228576.

a093562 n k = a093562_tabl !! n !! k
a093562_row n = a093562_tabl !! n
a093562_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [5, 1]
-- Reinhard Zumkeller, Aug 31 2014

from math import comb, isqrt
def A093562(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+(r-a<<2))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m) = F(5;n-m, m) for 0<= m <= n, otherwise 0, with F(5;0, 0)=1, F(5;n, 0)=5 if n>=1 and F(5;n, m):=(5*n+m)*binomial(n+m-1, m-1)/m if m>=1.
G.f. column m (without leading zeros): (1+4*x)/(1-x)^(m+1), m>=0.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=5 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
T(n, k) = C(n, k) + 4*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)*(5 + 11*x + 7*x^2/2! + x^3/3!) = 5 + 16*x + 34*x^2/2! + 60*x^3/3! + 95*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

A254142 a(n) = (9n+10)binomial(n+9,9)/10.

Original entry on oeis.org

1, 19, 154, 814, 3289, 11011, 32032, 83512, 199342, 442442, 923780, 1830764, 3468374, 6317234, 11113784, 18958808, 31461815, 50930165, 80613390, 125014890, 190285095, 284712285, 419329560, 608658960, 871616460, 1232604516, 1722822024, 2381824984

Offset: 0

Bruno Berselli, Jan 26 2015

Partial sums of A056003.

If n is of the form 8*k+2*(-1)^k-1 or 8*k+2*(-1)^k-2 then a(n) is odd.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000581, A001287, A056003.

Cf. sequences of the type (k*n+k+1)*binomial(n+k,k)/(k+1): A000217 (k=1), A000330 (k=2), A001296 (k=3), A034263 (k=4), A051946 (k=5), A034265 (k=6), A034266 (k=7), A056122 (k=8), this sequence (k=9).

List([0..30], n-> (9*n+10)*Binomial(n+9,9)/10); # G. C. Greubel, Aug 28 2019

[(9*n+10)*Binomial(n+9,9)/10: n in [0..30]];

seq((9*n+10)*binomial(n+9,9)/10, n=0..30); # G. C. Greubel, Aug 28 2019

Table[(9n+10)Binomial[n+9, 9]/10, {n, 0, 30}]

vector(30, n, n--; (9*n+10)*binomial(n+9, 9)/10)

[(9*n+10)*binomial(n+9,9)/10 for n in (0..30)]

G.f.: (1 + 8*x)/(1-x)^11.
a(n) = Sum_{i=0..n} (i+1)*A000581(i+8).
a(n+1) = 8*A001287(n+10) + A001287(n+11).

A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

Original entry on oeis.org

1, 7, 1, 18, 8, 1, 34, 26, 9, 1, 55, 60, 35, 10, 1, 81, 115, 95, 45, 11, 1, 112, 196, 210, 140, 56, 12, 1, 148, 308, 406, 350, 196, 68, 13, 1, 189, 456, 714, 756, 546, 264, 81, 14, 1, 235, 645, 1170, 1470, 1302, 810, 345, 95, 15, 1, 286, 880, 1815, 2640, 2772, 2112, 1155, 440, 110, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

The leftmost column contains the heptagonal numbers A000566.
The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.
Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.

			First few rows of the triangle are:
  1;
  7, 1;
  18, 8, 1;
  34, 26, 9, 1;
  55, 60, 35, 10, 1;
  81, 115, 95, 45, 11, 1;
  112, 196, 210, 140, 56, 12, 1;
Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1966, p. 189.

Cf. A000566, A002413, A002418, A027800, A051946, A050484.

Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.

A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n,k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1,k-1)+procname(n-1,k) ; fi; end: seq(seq(A125234(n,k),k=0..n-1),n=1..16) ; # R. J. Mathar, Sep 09 2009

A000566[n_] := n(5n-3)/2;
T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.

Edited and extended by R. J. Mathar, Sep 09 2009

A143130 Triangle read by rows, A000012 * (A000332 * 0^(n-k)) * A000012, 0<=k<=n.

Original entry on oeis.org

1, 6, 5, 21, 20, 15, 56, 55, 50, 35, 126, 125, 120, 105, 70, 252, 251, 246, 231, 196, 126, 462, 461, 456, 441, 406, 336, 210, 792, 791, 786, 771, 736, 666, 540, 330, 1287, 1286, 1281, 1266, 1231, 1161, 1035, 825, 495, 2002, 2001, 1996, 1981, 1946, 1876, 1750

Offset: 1

Gary W. Adamson, Jul 27 2008

Row sums = A051946: (1, 11, 56, 196, 546, 1302,...).

			First few rows of the triangle =
1;
6, 5;
21, 20, 15;
56, 55, 50, 35;
126, 125, 120, 105, 70;
252, 251, 246, 231, 196, 126;
462, 461, 456, 441, 406, 336, 210;
792, 791, 786, 771, 736, 666, 540, 330;
...

Cf. A000332, A051946.

Triangle read by rows, A000012 * (A000332 * 0^(n-k)) * A000012, 0<=k<=n; where A000332 starts (1, 5, 15, 35, 70,...).

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A050484 Partial sums of A051946.

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A093562 (5,1) Pascal triangle.

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A254142 a(n) = (9*n+10)*binomial(n+9,9)/10.

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A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

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A143130 Triangle read by rows, A000012 * (A000332 * 0^(n-k)) * A000012, 0<=k<=n.

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A254142 a(n) = (9n+10)binomial(n+9,9)/10.