A212500 -id:A212500 - cp's OEIS frontend

A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

1, 1, 3, 1, 10, 5, 1, 21, 35, 7, 1, 36, 126, 84, 9, 1, 55, 330, 462, 165, 11, 1, 78, 715, 1716, 1287, 286, 13, 1, 105, 1365, 5005, 6435, 3003, 455, 15, 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17, 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19, 1, 210, 5985, 54264, 203490, 352716, 293930, 116280, 20349, 1330, 21

Offset: 0

Wolfdieter Lang, Jan 23 2004

The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.
Elements have the same parity as those of Pascal's triangle.
All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - Vladimir Shevelev and Peter J. C. Moses, May 22 2012
Row reverse of A103327. - Peter Bala, Jul 29 2013
The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - Wolfdieter Lang, Oct 12 2017

			Triangle a(n, m) begins:
n\m  0   1    2     3      4      5      6      7     8    9  10 ...
0:   1
1:   1   3
2:   1  10    5
3:   1  21   35     7
4:   1  36  126    84      9
5:   1  55  330   462    165     11
6:   1  78  715  1716   1287    286     13
7:   1 105 1365  5005   6435   3003    455     15
8:   1 136 2380 12376  24310  19448   6188    680    17
9:   1 171 3876 27132  75582  92378  50388  11628   969   19
10:  1 210 5985 54264 203490 352716 293930 116280 20349 1330  21
... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
From _Peter Bala_, Jan 30 2022: (Start)
(1/2)*(N^2 + N) = Sum_{j = 1..N} j.
(1/2)*(N^2 + N)^3 = Sum_{j = 1..N} j^3 + 3*Sum_{j = 1..N} j^5.
(1/2)*(N^2 + N)^5 = Sum_{j = 1..N} j^5 + 10*Sum_{j = 1..N} j^7 + 5*Sum_{j = 1..N} j^9.
(1/2)*(N^2 + N)^7 = Sum_{j = 1..N} j^7 + 21*Sum_{j = 1..N} j^9 + 35*Sum_{j = 1..N} j^11 + 7*Sum_{j = 1..N} j^13. (End)

A. M. Yaglom and I. M. Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for pi. Uspekhi Matem. Nauk, VIII (1953), 181-187(in Russian).

Indranil Ghosh, Rows 0..120 of triangle, flattened
Wolfdieter Lang, First 9 rows.
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
V. Shevelev and P. Moses, Tangent power sums and their applications, arXiv:1207.0404 [math.NT], 2012-2014.

Cf. A212500, A038754. A000302 (row sums), A085478, A103327 (row reverse), A048854, A103328.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k) ))); # G. C. Greubel, Aug 01 2019

[[Binomial(2*n+1,2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Aug 01 2019

f := (x, t) -> cosh(sqrt(x)*t)*sinh(t); seq(seq(coeff(((2*n+1)!*coeff(series(f(x,t),t,2*n+2),t,2*n+1)),x,k),k=0..n),n=0..9); # Peter Luschny, Jul 29 2013

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2T[n-1, k] + 2T[n-1, k-1] + 2T[n-2, k-1] - T[n-2, k] - T[n-2, k-2]; T[0, 0] = T[1, 0] = 1; T[1, 1] = 3; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jul 29 2018, after Philippe Deléham *)
Table[Binomial[2*n+1, 2*k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,k) = binomial(2*n+1, 2*k); \\ G. C. Greubel, Aug 01 2019

from math import comb, isqrt
def A091042(n): return comb((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))<<1|1,n-comb(r+1,2)<<1) # Chai Wah Wu, Apr 30 2025

[[binomial(2*n+1, 2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, m) = binomial(2*n+1, 2*m) = A007318(2*n+1, 2*m), n >= m >= 0, otherwise 0.
From Peter Bala, Jul 29 2013: (Start)
E.g.f.: sinh(t)*cosh(sqrt(x)*t) = t + (1 + 3*x)*t^3/3! + (1 + 10*x + 5*x^2)*t^5/5! + (1 + 21*x + 35*x^2 + 7*x^3)*t^7/7! + ....
O.g.f.: A(x,t) = (1 + (x - 1)*t)/( (1 + (x - 1)*t)^2 - 4*t*x ) = 1 + (1 + 3*x)*t + (1 + 10*x + 5*x^2)*t^2 + ...
The function A( x/(x + 4), t*(x + 4)/4 ) = 1 + (1 + x)*t + (1 + 3*x + x^2)*t^2 + ... is the o.g.f. for A085478.
O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k)*x^k )/(1 - x)^(2*n).
n-th row polynomial R(n,x) = (1/2)*( (1 + sqrt(x))^(2*n+1) - (sqrt(x) - 1)^(2*n+1) ).
Row sums A000302. (End)
T(n, k) = 2*T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k) - T(n-2,k-2) with T(0,0)=T(1,0)=1, T(1,1)=3, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 26 2013
From Peter Bala, Jan 31 2022: (Start)
Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 0: (1/2)*(N^2 + N)^(2*n+1) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n)* S(4*n+1,N). Some examples are given below. (End)

A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).

Original entry on oeis.org

1, 6, 11, 106, 201, 2022, 3843, 38794, 73745, 744646, 1415547, 14293930, 27172313, 274381478, 521590643, 5266936010, 10012281377, 101102361990, 192192442603, 1940727511786, 3689262580969, 37253563629926, 70817864678883, 715107089849866

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

In general for all z, given a sequence of the form: a(n) is the difference between multiples of 2z+1 with even and odd digit sum in base 2z in interval [0,(2z)^n); then a(n) = (a(n+1) + a(n-1))/2 when n is even. The equation applies here where z=3. - Bob Selcoe, Jun 10 2014

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (0,21,0,-35,0,7).

Cf. A038754, A212500, A091042.

LinearRecurrence[{0, 21, 0, -35, 0, 7}, {1, 6, 11, 106, 201, 2022}, 24] (* Bruno Berselli, May 22 2012 *)

For n>=7, a(n) = 21*a(n-2)-35*a(n-4)+7*a(n-6).
G.f.: x*(1+6*x-10*x^2-20*x^3+5*x^4+6*x^5)/(1-21*x^2+35*x^4-7*x^6). [Bruno Berselli, May 22 2012]
a(n) = 2a(n-1) - a(n-2) when n is odd; a(n) = (a(n+1) + a(n-1))/2 when n is even. - Bob Selcoe, Jun 10 2014

A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

Original entry on oeis.org

9, 65, 201, 449, 841, 1409, 2185, 3201, 4489, 6081, 8009, 10305, 13001, 16129, 19721, 23809, 28425, 33601, 39369, 45761, 52809, 60545, 69001, 78209, 88201, 99009, 110665, 123201, 136649, 151041, 166409, 182785, 200201, 218689, 238281, 259009, 280905, 304001

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 32*n^5).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A038754, A084990, A091042, A212500, A212592.

[(16/3)*(n+1)*n*(n-1)+8*n^2+1: n in [1..40]]; // Vincenzo Librandi, Dec 01 2015

LinearRecurrence[{4, -6, 4, -1}, {9, 65, 201, 449}, 40] (* Vincenzo Librandi, Dec 01 2015 *)
CoefficientList[Series[x (9+29x-5x^2-x^3)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Mar 29 2023 *)

a(n)=16*(n+1)*n*(n-1)/3+8*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

Vec(x*(9+29*x-5*x^2-x^3)/(1-x)^4 + O(x^100)) \\ Colin Barker, Nov 30 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^5(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).

G.f.: x*(9+29*x-5*x^2-x^3) / (1-x)^4. - Colin Barker, Nov 30 2015

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

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

Cf. A212500

b[n_]:=n;c[n_]:=3n-1;
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}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

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(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A213831 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 3n-5+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 4, 24, 19, 7, 58, 51, 31, 10, 115, 106, 78, 43, 13, 201, 190, 154, 105, 55, 16, 322, 309, 265, 202, 132, 67, 19, 484, 469, 417, 340, 250, 159, 79, 22, 693, 676, 616, 525, 415, 298, 186, 91, 25, 955, 936, 868, 763, 633

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213832.
Antidiagonal sums: A212560.
row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.
Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.
Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.
For a guide to related arrays, see A212500.

			1....7....24....58....115
4....19...51....106...190
7....31...78....154...265
10...43...105...202...340
13...55...132...250...415

Cf. A212500

b[n_]:=2n-1;c[n_]:=3n-2;
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}] (* A213831 *)
Table[t[n,n],{n,1,40}] (* A213832 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A212560 *)

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*((3*n-2) + 3*x - (3*n-5)*x^2) and g(x) = (1-x)^4.
Northwest corner (the array is read by falling antidiagonals):

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 3, 24, 17, 5, 58, 48, 27, 7, 115, 102, 72, 37, 9, 201, 185, 146, 96, 47, 11, 322, 303, 255, 190, 120, 57, 13, 484, 462, 405, 325, 234, 144, 67, 15, 693, 668, 602, 507, 395, 278, 168, 77, 17, 955, 927, 852, 742, 609, 465

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A103748.
Antidiagonal sums: A213834.
Row 1, (1,3,5,7,...)**(1,3,5,7,...): A081436.
Row 2, (1,3,5,7,...)**(3,5,7,9,...): A144640.
Row 3, (1,3,5,7,...)**(5,7,9,11,...): (2*k^3 + 11*k^2 - 3*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....7....24....58....115
3....17...48....102...185
5....27...72....146...255
7....37...96....190...325
9....47...120...234...395
11...57...144...278...465

Clark Kimberling, Antidiagonals n = 1..12, flattened

Cf. A212500.

b[n_]:=3n-2;c[n_]:=2n-1;
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}] (* A213833 *)
Table[t[n,n],{n,1,40}] (* A130748 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213834 *)

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*((2*n-1) + (2*n+1)*x - (4*n-6)*x^2) and g(x) = (1-x)^4.

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.

A213841 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 4n-7+4*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 8, 5, 29, 24, 9, 72, 65, 40, 13, 145, 136, 101, 56, 17, 256, 245, 200, 137, 72, 21, 413, 400, 345, 264, 173, 88, 25, 624, 609, 544, 445, 328, 209, 104, 29, 897, 880, 805, 688, 545, 392, 245, 120, 33, 1240, 1221, 1136, 1001

Offset: 1

Clark Kimberling, Jul 05 2012

Principal diagonal: A213842.
Antidiagonal sums: A213843.
Row 1, (1,5,9,13,...)**(1,3,5,7,...): A100178.
Row 2, (1,5,9,13,...)**(3,5,7,9,...): (4*k^3 + 9*k^2 + 2*k)/3.
Row 3, (1,5,9,13,...)**(5,7,9,11,...): (4*k^3 + 21*k^2 + 2*k)/3.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....8....29....72....145
5....24...65....136...245
9....40...101...200...345
13...56...137...264...445
17...72...173...328...545
21...88...209...392...645

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

Cf. A212500.

b[n_]:=2n-1;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}] (* A213841 *)
Table[t[n,n],{n,1,40}] (* A213842 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213843 *)

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*x - (4*n-7)*x^2) and g(x) = (1-x)^4.

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

Original entry on oeis.org

18, 340, 2022, 7400, 20602, 48060, 99022, 186064, 325602, 538404, 850102, 1291704, 1900106, 2718604, 3797406, 5194144, 6974386, 9212148, 11990406, 15401608, 19548186, 24543068, 30510190, 37585008, 45915010, 55660228, 66993750, 80102232, 95186410, 112461612

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 64*n^6).

Colin Barker, Table of n, a(n) for n = 1..1000
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A038754, A084990, A091042, A212500, A212592, A212592, A212592, A212668.

Vec(2*x*(9+116*x+126*x^2+4*x^3+x^4)/(1-x)^6 + O(x^50)) \\ Colin Barker, Dec 01 2015

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^6(Pi*i/(2*n+1)).

G.f.: 2*x*(9+116*x+126*x^2+4*x^3+x^4) / (1-x)^6. - Colin Barker, Dec 01 2015

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 4, 18, 15, 7, 40, 36, 24, 10, 75, 70, 54, 33, 13, 126, 120, 100, 72, 42, 16, 196, 189, 165, 130, 90, 51, 19, 288, 280, 252, 210, 160, 108, 60, 22, 405, 396, 364, 315, 255, 190, 126, 69, 25, 550, 540, 504, 448, 378, 300

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A172073.
Antidiagonal sums: A002419.
Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411.
Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414.
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2.
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
4....15...36...70....120...189
7....24...54...100...165...252
10...33...72...130...210...315
13...42...90...160...255...378

Clark Kimberling, Antidiagonals n = 1..45, flattened

Cf. A212500.

b[n_]:=n;c[n_]:=3n-2;
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}] (* A213761 *)
Table[t[n,n],{n,1,40}] (* A172073 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A002419 *)

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*(3*n - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4.

cp's OEIS Frontend

A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

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A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).

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A212668 a(n) = (16/3)*(n+1)*n*(n-1) + 8*n^2 + 1.

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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

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A213831 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

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A213833 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

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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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A212668 a(n) = (16/3)(n+1)n(n-1) + 8n^2 + 1.

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

A213831 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 3n-5+3*h, n>=1, h>=1, and ** = convolution.

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, 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.

A213841 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 4n-7+4*h, n>=1, h>=1, and ** = convolution.

A212669 a(n) = 2/15 * (32n^5 + 80n^4 + 40n^3 - 20n^2 + 3*n).

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.