A101338 -id:A101338 - cp's OEIS frontend

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544

Cf. A000217, A001263, A101338.

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
T(10,m) = A062786(m+1).
T(11,m) = A069125(m+1).
T(12,m) = A003154(m+1).
T(13,m) = A069126(m+1).
T(14,m) = A069127(m+1).
T(15,m) = A069128(m+1).
T(16,m) = A069129(m+1).
T(17,m) = A069130(m+1).
T(18,m) = A069131(m+1).
T(19,m) = A069132(m+1).
T(20,m) = A069133(m+1).
T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Edited by R. J. Mathar, Oct 21 2009

A154322 a(n) = 1 + n + binomial(n+3,5).

Original entry on oeis.org

1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, 2014, 3016, 4382, 6203, 8584, 11645, 15522, 20368, 26354, 33670, 42526, 53153, 65804, 80755, 98306, 118782, 142534, 169940, 201406, 237367, 278288, 324665, 377026, 435932, 501978, 575794, 658046, 749437, 850708

Offset: 0

Paul Barry, Jan 07 2009

Row sums of number triangle A113582.
It appears that the sequence is the pairwise sum of terms in A101338 and A000389 with offsets as follows:
1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, ... =
1, 2, 4,  9, 20, 41,  77, 134, 219, 340,  506, ... +
0, 0, 0,  1,  6, 21,  56, 126, 252, 462,  792, ...
- Gary W. Adamson, Oct 08 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A101338, A113582.

[1+n+Binomial(n+3,5) : n in [0..50]]; // Wesley Ivan Hurt, Oct 08 2015

I:=[1,2,4,10,26,62]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 09 2015

A154322:=n->1+n+binomial(n+3,5): seq(A154322(n), n=0..50); # Wesley Ivan Hurt, Oct 08 2015

CoefficientList[Series[(1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - x)^6, {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 08 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 2, 4, 10, 26, 62}, 50] (* Vincenzo Librandi, Oct 09 2015 *)
 Table[ (n + 1)*(n^4 + 4*n^3 + n^2 - 6*n + 120)/120 , {n, 0, 25}] (* G. C. Greubel, Sep 10 2016 *)
Table[1+n+Binomial[n+3,5],{n,0,40}] (* Harvey P. Dale, Jan 19 2023 *)

Vec((1-4*x+7*x^2-4*x^3+x^4)/(1-x)^6 + O(x^100)) \\ Altug Alkan, Oct 18 2015

G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1-x)^6;
a(n) = n + 1 + Sum_{k=0..n} binomial(k+1,2) * binomial(n-k+1,2).
a(n) = (n+1)*(n^4 +4*n^3 +n^2 -6*n +120)/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) for n>5. - Wesley Ivan Hurt, Oct 08 2015
E.g.f.: (1/120)*(120 + 120*x + 60*x^2 + 60*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Sep 10 2016

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A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

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A154322 a(n) = 1 + n + binomial(n+3,5).

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cp's OEIS Frontend

A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

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Author

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Maple

Mathematica

PARI

Formula

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A154322 a(n) = 1 + n + binomial(n+3,5).

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Author

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Magma

Magma

Maple

Mathematica

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Formula

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.