A064808 -id:A064808 - cp's OEIS frontend

A076110 Triangle (read by rows) in which the n-th row contains first n terms of an arithmetic progression with first term 1 and common difference (n-1).

1, 1, 2, 1, 3, 5, 1, 4, 7, 10, 1, 5, 9, 13, 17, 1, 6, 11, 16, 21, 26, 1, 7, 13, 19, 25, 31, 37, 1, 8, 15, 22, 29, 36, 43, 50, 1, 9, 17, 25, 33, 41, 49, 57, 65, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122

Offset: 1

Amarnath Murthy, Oct 09 2002

Leading diagonal contains n^2 + 1 (A002522).

Sum of the n-th row is (n+1)(n^2+2)/2 (A064808).

			1;
1, 2;
1, 3, 5;
1, 4, 7, 10;
1, 5, 9, 13, 17;
1, 6, 11, 16, 21, 26;
1, 7, 13, 19, 25, 31, 37; ...

Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Cf. A002522, A064808, A076111 (row products), A079904.

Flat(List([1..12],n->List([1..n],k->1+(n-1)*(k-1)))); # Muniru A Asiru, Dec 05 2018

/* As triangle */ [[1+(n-1)*(k-1): k in [1..n]]: n in [1.. 12]]; // Vincenzo Librandi, Dec 05 2018

T:= (n,k) -> 1+(n-1)*(k-1):for n from 1 to 10 do seq(T(n,k),k=1..n) od; # Robert Israel, Dec 04 2018

T[n_, k_] := 1 + (n-1) * (k-1); Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* Amiram Eldar, Dec 04 2018 *)

A076110(n) = L(n) with L=seq(seq(n*k+1, k = 0..n), n = 0..+inf). - Yalcin Aktar, Jul 14 2009
From Robert Israel, Dec 04 2018: (Start)
T(n,k) = 1 + (n-1)*(k-1).
G.f. as triangle: (1-x-x*y+2*x^2*y+2*x^2*y^2-3*x^3*y^2)*x*y/((1-x)^2*(1-x*y)^3).
G.f. as sequence: x/(1-x) + Sum_{m>=0} (-m*(m+1)*x^((m^2+3*m+4)/2) + (1+m*(m+1))*x^((m^2+3*m+6)/2))/(1-x)^2.
(End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A081113 Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.

Original entry on oeis.org

1, 4, 17, 68, 259, 950, 3387, 11814, 40503, 136946, 457795, 1515926, 4979777, 16246924, 52694573, 170028792, 546148863, 1747255194, 5569898331, 17698806798, 56076828573, 177208108824, 558658899825, 1757365514652

Offset: 1

David Scambler, Apr 16 2003

a(n) = number of sequences (a_1,a_2,...,a_n) with 1<=a_i<=n for all i and |a_(i+1)-a_(i)|<=1 for 1<=i<=n-1. For n=2 the sequences are 11, 12, 21, 22. - David Callan, Oct 24 2004

Simon Plouffe proposes the ordinary generating function A(x) (for offset zero) in the implicit form 3-10*x+12*x^2+(-4+30*x+54*x^3-72*x^2)*A(x)+(81*x^4+54*x^2+1-12*x-108*x^3)*A(x)^2 = 0 which delivers at least the first 200 terms (i.e., as far as tested) correctly. - David Scambler, R. J. Mathar, Jan 06 2011

			For n=2 the 4 paths are (0,0)->(0,1); (0,0)->(1,1); (1,0)->(0,1); (1,0)->(1,1).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Simon Plouffe, OEIS conjectured formulas.
D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I , arXiv:1612.08697 [math.CO], 2016-2017.

Cf. A005773 (paths which begin at a corner), diagonal of A296449.

A026300 := proc(n,k) add( binomial(n,2*i+n-k)*(binomial(2*i+n-k,i) -binomial(2*i+n-k,i-1)), i=0..floor(k/2)) ; end proc:
A081113 := proc(n) add(k*(n-k+1)*A026300(n-1,k-1),k=1..n) ; end proc:
seq(A081113(n),n=1..20) ;
# R. J. Mathar, Jun 09 2010

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}] (* from A026300 *); f[n_] := Sum[ k(n - k + 1)t[n - 1, k - 1], {k, n}]; Array[f, 24]

a(n) = Sum_{k=1..n} k*(n-k+1)*M(n-1, k-1) where k*(n-k+1) is the triangular view of A003991 and M() is the Motzkin triangle A026300.
Conjecture: g.f.(x)=z*A064808(z), where z=x*A001006(x) and A...(x) are the corresponding generating functions. - R. J. Mathar, Jul 07 2009
Conjecture from WolframAlpha (verified for 1<=n<=180): (n+3)*a(n+4) = 27*n*a(n) +27*a(n+1) -9*(2*n+5)*a(n+2) +(8*n+21)*a(n+3). - Alexander R. Povolotsky, Jan 04 2011
Shorter recurrence: (n-1)*(2*n-7)*a(n) = (10*n^2-39*n+23)*a(n-1) - 3*(2*n^2-n-9)*a(n-2) - 9*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 28 2012
a(n) ~ 3^(n-1)*n*(1-4/(sqrt(3*Pi*n))). - Vaclav Kotesovec, Oct 28 2012
a(n) = (n+2)*3^(n-2)+2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corollary 2.8] - R. J. Mathar, Dec 13 2017

A076111 Product of terms in n-th row of A076110.

Original entry on oeis.org

1, 2, 15, 280, 9945, 576576, 49579075, 5925744000, 939536222625, 190787784140800, 48279601331512551, 14894665739501184000, 5502449072258318805625, 2397950328737212204032000

Offset: 0

Amarnath Murthy, Oct 09 2002

G. C. Greubel, Table of n, a(n) for n = 0..230

Cf. A002522, A064808, A076110.

List([0..15], n-> Product([1..n], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020

[1] cat [&*[j*n+1: j in [1..n]]: n in [1..15]]; // G. C. Greubel, Mar 04 2020

seq( mul(j*n+1, j=1..n), n=0..15); # G. C. Greubel, Mar 04 2020

Table[Product[j*n+1, {j,n}], {n,0,15}] (* G. C. Greubel, Mar 04 2020 *)

A076111(n):=prod(1+n*k,k,1,n)$
makelist(A076111(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

vector(16, n, my(m=n-1); prod(j=1,m, j*m+1)) \\ G. C. Greubel, Mar 04 2020

[product(j*n+1 for j in (1..n)) for n in (0..15)] # G. C. Greubel, Mar 04 2020

a(n) = Prod_{k=1..n} (1+n*k). - Yalcin Aktar, Jul 14 2009
a(n) = n^n * Pochhammer(n, 1 + 1/n). - G. C. Greubel, Mar 04 2020
a(n) = A092985(n)*(n^2+1). - R. J. Mathar, Mar 30 2023

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A131676 a(n) = (Product_{i=1..6} n^i+i) / 6!.

Original entry on oeis.org

1, 7, 14245, 28405300, 9191136045, 886286703456, 38188743738145, 932714257963020, 14966184483875625, 173860405001195185, 1563721100613810061, 11427034989921521488, 70319024498214551605, 374482754394635213250, 1763001772206469563945, 7462412915610398239816

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

Index entries for linear recurrences with constant coefficients, signature (22, -231, 1540, -7315, 26334, -74613, 170544, -319770, 497420, -646646, 705432, -646646, 497420, -319770, 170544, -74613, 26334, -7315, 1540, -231, 22, -1).

Cf. A131685.

Table[Product[ n^i + i, {i, 1, 6}]/6!, {n, 0, 15}] (* Michael De Vlieger, Jan 03 2016 *)

A131676(n, k=6)=prod(i=1, k, (n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

G.f.: (1 - 15*x + 14322*x^2 + 28091987*x^3 + 8569506575*x^4 + 690621422337*x^5 + 20769948618958*x^6 + 283347184706283*x^7 + 1969675285865562*x^8 + 7493939424807955*x^9 + 16292973927985678*x^10 + 20712738704664489*x^11 + 15498276638623618*x^12 + 6765765599122915*x^13 + 1679542499740050*x^14 + 226176197184209*x^15 + 15278037714093*x^16 + 454493699352*x^17 + 4732512736*x^18 + 10869320*x^19 + 1575*x^20)/(1 - x)^22. - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131679 a(n) = (Product_{i=1..9} n^i+i) / 9!.

Original entry on oeis.org

1, 10, 524816325, 15995379784360900, 5136081211768056707885, 104827108835105429096703456, 359044402823940369662885183425, 354548318931625565271233374406000, 140230322081790179721500725877795625, 27516367648544953143193233240569070880, 3102623679344954347223585172112606310061

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Table[Product[ n^i + i, {i, 1, 9}]/9!, {n, 0, 10}] (* Michael De Vlieger, Jan 03 2016 *)

A131679(n,k=9)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A144693 Triangle read by rows, A000012 * (3A144328 - 2A000012), where A000012 means a lower triangular matrix of all 1's.

Original entry on oeis.org

1, 2, 1, 3, 2, 4, 4, 3, 8, 7, 5, 4, 12, 14, 10, 6, 5, 16, 21, 20, 13, 7, 6, 20, 28, 30, 26, 16, 8, 7, 24, 35, 40, 39, 32, 19, 9, 8, 28, 42, 50, 52, 48, 38, 22, 10, 9, 32, 49, 60, 65, 64, 57, 44, 25, 11, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28

Offset: 1

Gary W. Adamson, Sep 19 2008

			Partial sums by columns of the triangle (3*A144328 - 2*A000012):
  1;
  1, 1;
  1, 1, 4;
  1, 1, 4, 7;
  1, 1, 4, 7, 10;
  ...
First few rows of the triangle:
  1;
  2, 1
  3, 2,  4;
  4, 3,  8,  7;
  5, 4, 12, 14, 10;
  6, 5, 16, 21, 20, 13;
  7, 6, 20, 28, 30, 26, 16;
  8, 7, 24, 35, 40, 39, 32, 19;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000012, A064808, A144328.

A144693:= func< n,k | k eq 1 select n else (3*k-5)*(n-k+1) >;
[A144693(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 19 2021

T[n_, k_]:= (3*k -5 +3*Boole[k==1])*(n-k+1);
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 19 2021 *)

def A144693(n,k): return (3*k -5 +3*bool(k==1))*(n-k+1)
flatten([[A144693(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 19 2021

Sum_{k=1..n} T(n, k) = A064808(n).

T(n, k) = (3*k -5 +3*[k=1])*(n-k+1). - G. C. Greubel, Oct 19 2021

A373170 Main diagonal of A373169.

Original entry on oeis.org

1, 3, 9, 22, 45, 81, 43, 24, 288, 118, 138, 585, 184, 918, 288, 232, 294, 351, 559, 2586, 1179, 3586, 117, 72, 3913, 2949, 5949, 1585, 7239, 8811, 3595, 3789, 3537, 5758, 1968, 18, 6454, 4152, 4374, 1687, 549, 7794, 3922, 8313, 828, 2674, 4251, 2646, 5548, 3636, 3879, 799

Offset: 2

Paolo Xausa, May 28 2024

a(n) is the zeroless analog of the (n-1)-th n-gonal number.

Paolo Xausa, Table of n, a(n) for n = 2..10000

Cf. A064808, A004719, A373169.

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
A373170[n_] := Fold[noz[#2*(n-2) + 1 + #] &, 1, Range[n-2]];
Array[A373170, 100, 2]

a(n) = A373169(n,n-1).

A131677 a(n) = (Product_{i=1..7} n^i+i) / 7!.

Original entry on oeis.org

1, 8, 274725, 8903032600, 21521701559085, 9892478959203456, 1527238784041075105, 109733832449349303000, 4483781212288588835625, 118795734924428077310080, 2233888850312257843810061, 31811523551546985038211552, 359951182400070234774044725

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000

Table[Product[ n^i + i, {i, 1, 7}]/7!, {n, 0, 12}] (* Michael De Vlieger, Jan 03 2016 *)

A131677(n,k=7)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A131678 a(n) = (Product_{i=1..8} n^i+i) / 8!.

Original entry on oeis.org

1, 9, 9065925, 7310502643675, 176327300873583405, 483041091658815453456, 320648364425775841520065, 79074323113562613259765875, 9403175220694650942397475625, 639220975955961365494757841040, 27923612862792073359883606310061, 852385355738368243011331354210716, 19346552845649626158477975728463925

Offset: 0

Alexander R. Povolotsky, Sep 15 2007

nonn

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000

Table[Product[ n^i + i, {i, 1, 8}]/8!, {n, 0, 12}] (* Michael De Vlieger, Jan 03 2016 *)

A131678(n,k=8)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995(k=4), A131675 (k=5), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

Definition made explicit by M. F. Hasler, May 02 2015

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Robert G. Wilson v, Apr 27 2020

\c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
r\
_0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
_1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
_2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
_3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
_4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
_5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
_6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
_7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
_8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
_9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

P(r, c) = (r - 2)(c(c-1)/2) + c.

cp's OEIS Frontend

A076110 Triangle (read by rows) in which the n-th row contains first n terms of an arithmetic progression with first term 1 and common difference (n-1).

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A081113 Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.

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A076111 Product of terms in n-th row of A076110.

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A131676 a(n) = (Product_{i=1..6} n^i+i) / 6!.

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A131679 a(n) = (Product_{i=1..9} n^i+i) / 9!.

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A144693 Triangle read by rows, A000012 * (3*A144328 - 2*A000012), where A000012 means a lower triangular matrix of all 1's.

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A373170 Main diagonal of A373169.

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A144693 Triangle read by rows, A000012 * (3A144328 - 2A000012), where A000012 means a lower triangular matrix of all 1's.