A086645 -id:A086645 - cp's OEIS frontend

A156740 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 7, read by rows.

1, 1, 1, 1, 153, 1, 1, 4845, 4845, 1, 1, 74613, 2362745, 74613, 1, 1, 735471, 358664691, 358664691, 735471, 1, 1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1, 1, 30421755, 1056158828725, 160324910200455, 160324910200455, 1056158828725, 30421755, 1

Offset: 0

Roger L. Bagula, Feb 14 2009

			Triangle begins as:
  1;
  1,       1;
  1,     153,           1;
  1,    4845,        4845,            1;
  1,   74613,     2362745,        74613,           1;
  1,  735471,   358664691,    358664691,      735471,       1;
  1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A086645 (m=0), A156739 (m=6), this sequence (m=7), A156741 (m=8), A156742 (m=9).

Cf. A151614 (row sums).

A156740:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..7]]) ) >;
[A156740(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021

b[n_, k_]:= Binomial[2*n, 2*k];
T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

def A156740(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..7)) )
flatten([[A156740(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 7.

Sum_{k=0..n} T(n, k, 7) = A151614(n).

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021

A156741 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 8, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 190, 1, 1, 7315, 7315, 1, 1, 134596, 5181946, 134596, 1, 1, 1562275, 1106715610, 1106715610, 1562275, 1, 1, 13123110, 107904771975, 1985447804340, 107904771975, 13123110, 1, 1, 86493225, 5974000557525, 1275875833357125, 1275875833357125, 5974000557525, 86493225, 1

Offset: 0

Roger L. Bagula, Feb 14 2009

			Triangle begins as:
  1;
  1,        1;
  1,      190,            1;
  1,     7315,         7315,             1;
  1,   134596,      5181946,        134596,            1;
  1,  1562275,   1106715610,    1106715610,      1562275,        1;
  1, 13123110, 107904771975, 1985447804340, 107904771975, 13123110, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), this sequence (m=8), A156742 (m=9).

Cf. A151709 (row sums).

A156741:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..8]]) ) >;
[A156741(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021

b[n_, k_]:= Binomial[2*n, 2*k];
T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
Table[T[n, k, 8], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

def A156741(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..8)) )
flatten([[A156741(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 8.

Sum_{k=0..n} T(n, k) = A151709(n).

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021

A201461 Triangle read by rows: n-th row (n>=0) gives coefficients of the polynomial ((x+1)^(2^n) + (x-1)^(2^n))/2.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 28, 70, 28, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 496, 35960, 906192, 10518300, 64512240, 225792840, 471435600, 601080390, 471435600, 225792840, 64512240, 10518300, 906192, 35960, 496, 1

Offset: 0

N. J. A. Sloane, Dec 01 2011

Wanted: reference for the fact that these polynomials are irreducible. Washington, Cyclotomic Fields, perhaps?
The algorithm r(n) = (1/2)*(r(n-1) + A/r(n-1)), starting with r(0) = A, used for approximating sqrt(A), which is known as the Babylonian method or Hero's method after the first-century Greek mathematician Hero of Alexandria and which can be derived from Newton's method, generates fractions beginning with (A+1)/2, (A^2 + 6*A + 1)/(4*(A+1)), (A^4 + 28*A^3 + 70*A^2 + 28*A + 1)/(8*(A+1)*(A^2 + 6*A + 1)), ... This is p(n,sqrt(A))/(2^n*Product_{k=1..n-1} p(k,sqrt(A))) with the given polynomial p(n,x) = ((x+1)^(2^n) + (x-1)^(2^n))/2. - Martin Renner, Jan 11 2017
The quadratic coefficient of this polynomial is A006516(n), the even-indexed coefficients are binomial(2^n,2*k) or A086645(2^(n-1),k) for 0 <= k <= 2^(n-1), in each row the maximum central coefficient for n>=2 is A037293(n) or A000984(2^(n-1)). - Martin Renner, Jan 14 2017
T(n,k) and A281122 are a bisection of row 2^n of Pascal's triangle A007318. - Martin Renner, Jan 15 2017
For nonnegative real x, sqrt(x) = (2*x/(1 + x)) * (2*(1 + x)^2/(1 + 6*x + x^2)) * (2*(1 + 6*x + x^2)^2/(1 + 28*x + 70*x^2 + 28*x^3 + x^4)) * .... See Bauer. - Peter Bala, Jan 18 2022

			The first few polynomials are:
1,
x^2 + 1,
x^4 + 6*x^2 + 1,
x^8 + 28*x^6 + 70*x^4 + 28*x^2 + 1,
x^16 + 120*x^14 + 1820*x^12 + 8008*x^10 + 12870*x^8 + 8008*x^6 + 1820*x^4 + 120*x^2 + 1.
The triangle of coefficients begins:
[0] [1]
[1] [1, 0, 1]
[2] [1, 0, 6, 0, 1]
[3] [1, 0, 28, 0, 70, 0, 28, 0, 1]
[4] [1, 0, 120, 0, 1820, 0, 8008, 0, 12870, 0, 8008, 0, 1820, 0, 120, 0, 1]
The triangle of nonzero coefficients begins:
[0] 1
[1] 1, 1
[2] 1, 6, 1
[3] 1, 28, 70, 28, 1
[4] 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1
[5] 1, 496, 35960, 906192, 10518300, 64512240, 225792840, 471435600, 601080390, 471435600, 225792840, 64512240, 10518300, 906192, 35960, 496, 1
...

Indranil Ghosh, Rows 0..11, flattened
F. L. Bauer, Letters to the editor: An Infinite Product for Square-Rooting with Cubic Convergence, The Mathematical Intelligencer, Vol. 20, Issue 1, (1998), 12-14.

Cf. A000984, A006516, A007318, A037293, A086645, A281122.

Flatten[Table[Binomial[2^n,2k],{n,0,6},{k,0,2^(n-1)}]] (* Indranil Ghosh, Feb 22 2017 *)

row(n) = my(v = Vec(((x+1)^(2^n)+(x-1)^(2^n))/2)); vector(#v\2 + 1, k, v[2*k-1]); \\ Michel Marcus, Jan 14 2017

T(n,k)=binomial(2^n,2*k);
for(n=0,5,for(k=0,2^(n-1),print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 15 2017

def A201461_polynomial(n): return expand(((x+1)^(2^n) + (x-1)^(2^n))/2)
for n in range(6): print(A201461_polynomial(n))
for n in range(6): print(A201461_polynomial(n).list()) # coefficients
for n in range(6): # depunched (not a mathematical operation)
    if n == 0: print([1])
    else: print(A201461_polynomial(n).list()[::2]) # Peter Luschny, Jan 11 2021

T(n,k) = binomial(2^n,2*k). - Joerg Arndt, Jan 15 2017

A109446 Binomial coefficients C(n,k) with n-k even, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 1, 5, 10, 1, 1, 15, 15, 1, 7, 35, 21, 1, 1, 28, 70, 28, 1, 9, 84, 126, 36, 1, 1, 45, 210, 210, 45, 1, 11, 165, 462, 330, 55, 1, 1, 66, 495, 924, 495, 66, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 15, 455, 3003, 6435, 5005

Offset: 0

Philippe Deléham, Aug 27 2005

Binomial(n,2(n-k-1)) is also the number of permutations avoiding both 123 and 132 with k descents, i.e., positions with w[i]>w[i+1]. - Lara Pudwell, Dec 19 2018

			Starred terms in Pascal's triangle (A007318), read by rows:
  1*;
  1,  1*;
  1*,  2,  1*;
  1,  3*,   3,  1*;
  1*,  4,  6*,   4,  1*;
  1,  5*,  10, 10*,   5,   1*;
  1*,  6, 15*,  20, 15*,    6,  1*;
  1,  7*,  21, 35*,  35,  21*,   7,  1*;
  1*,  8, 28*,  56, 70*,   56, 28*,   8, 1*;
  1,  9*,  36, 84*, 126, 126*,  84, 36*,  9, 1*;
Rows in A086645 (1; 1, 1; 1, 6, 1; ...) interspersed with rows in A103327 (1; 3, 1; 5, 10, 1; ...).
1; 1; 1, 1; 3, 1; 1, 6, 1; 5, 10, 1; 1, 15, 15, 1; 7, 35, 21, 1; ....

Alois P. Heinz, Rows n = 0..200, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

Cf. A109447. See A054142 for another version.

T:= (n, k)-> binomial(n, 2*k+irem(n, 2)):
seq(seq(T(n, k), k=0..floor(n/2)), n=0..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ EvenQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

A156739 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 6, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 120, 1, 1, 3060, 3060, 1, 1, 38760, 988380, 38760, 1, 1, 319770, 103285710, 103285710, 319770, 1, 1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1, 1, 9657700, 157843517260, 16494647553670, 16494647553670, 157843517260, 9657700, 1

Offset: 0

Roger L. Bagula, Feb 14 2009

			Triangle begins as:
  1;
  1,       1;
  1,     120,          1;
  1,    3060,       3060,           1;
  1,   38760,     988380,       38760,          1;
  1,  319770,  103285710,   103285710,     319770,       1;
  1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A086645 (m=0), this sequence (m=6), A156740 (m=7), A156741 (m=8), A156742 (m=9).

A156739:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..6]]) ) >;
[A156739(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 18 2021

b[n_, k_]:= Binomial[2*n, 2*k];
T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
Table[T[n, k, 6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 18 2021 *)

def A156739(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..6)) )
flatten([[A156739(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 18 2021

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 6.

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 18 2021

A156742 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 9, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 231, 1, 1, 10626, 10626, 1, 1, 230230, 10590580, 230230, 1, 1, 3108105, 3097744650, 3097744650, 3108105, 1, 1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1, 1, 225792840, 29367745734600, 8590065627370500, 8590065627370500, 29367745734600, 225792840, 1

Offset: 0

Roger L. Bagula, Feb 14 2009

			Triangle begins as:
  1;
  1,        1;
  1,      231,            1;
  1,    10626,        10626,             1;
  1,   230230,     10590580,        230230,            1;
  1,  3108105,   3097744650,    3097744650,      3108105,        1;
  1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1;

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), A156741 (m=8), this sequence (m=9).

A156742:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..9]]) ) >;
[A156742(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2021

T[n_, k_, m_]:= Round[Product[Binomial[2*(n+j), 2*(k+j)]/Binomial[2*(n-k+j), 2*j], {j,0,m}]];
Table[T[n, k, 9], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 19 2021 *)

def A156742(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..9)) )
flatten([[A156742(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2021

T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 9.

Definition corrected to give integral terms and edited by G. C. Greubel, Jun 19 2021

A177808 Triangle T(n,m) = binomial(4n, 4m), 0 <= m <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 70, 1, 1, 495, 495, 1, 1, 1820, 12870, 1820, 1, 1, 4845, 125970, 125970, 4845, 1, 1, 10626, 735471, 2704156, 735471, 10626, 1, 1, 20475, 3108105, 30421755, 30421755, 3108105, 20475, 1, 1, 35960, 10518300, 225792840, 601080390, 225792840, 10518300, 35960, 1, 1, 58905, 30260340, 1251677700, 7307872110, 7307872110, 1251677700, 30260340, 58905, 1

Offset: 0

Roger L. Bagula, Dec 13 2010

Row sums are A070775(n).

			1;
1, 1;
1, 70, 1;
1, 495, 495, 1;
1, 1820, 12870, 1820, 1;
1, 4845, 125970, 125970, 4845, 1;
1, 10626, 735471, 2704156, 735471, 10626, 1;
1, 20475, 3108105, 30421755, 30421755, 3108105, 20475, 1;
1, 35960, 10518300, 225792840, 601080390, 225792840, 10518300, 35960, 1;
1, 58905, 30260340, 1251677700, 7307872110, 7307872110, 1251677700,30260340, 58905, 1;
1, 91390, 76904685, 5586853480, 62852101650, 137846528820, 62852101650, 5586853480, 76904685, 91390, 1;

Cf. A086645, A034839, A070775.

A177808 := proc(n,m) binomial(4*n,4*m) ; end proc: # R. J. Mathar, Dec 13 2010

t[n_, m_] = Binomial[n, 4*m];
Table[Table[t[n, m], {m, 0, Floor[n/4]}], {n, 0, 40, 4}];
Flatten[%]

Right-left symmetric: T(n,m) = T(n,n-m).

A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 30, 30, 1, 1, 56, 140, 56, 1, 1, 90, 420, 420, 90, 1, 1, 132, 990, 1848, 990, 132, 1, 1, 182, 2002, 6006, 6006, 2002, 182, 1, 1, 240, 3640, 16016, 25740, 16016, 3640, 240, 1, 1, 306, 6120, 37128, 87516, 87516, 37128, 6120, 306, 1, 1, 380, 9690, 77520, 251940, 369512, 251940, 77520, 9690, 380, 1

Offset: 0

Peter Bala, Feb 22 2011

Given a subset S of the integers Z, Bhargava [1] has shown how to associate with S a generalized factorial function, denoted n!_S, sharing many properties of the classical factorial function n! (which corresponds to the choice S = Z). In particular, he shows that the generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) are always integral for any choice of S. Here we take S = {0,1,4,9,16,...}, the set of squares.
The associated generalized factorial function n!_S is given by the formula
n!S = Product{k=0..n} (n^2 - k^2), with the convention 0!S = 1. This should be compared with n! = Product{k=0..n} (n - k).
For n >= 1, n!_S = (2*n)!/2 = A002674(n).
Compare this triangle with A086645 and also A186430 - the generalized binomial coefficients for the set S of prime numbers {2,3,5,7,11,...}.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1....12.....1
.3..|..1....30....30.....1
.4..|..1....56...140....56.....1
.5..|..1....90...420...420....90.....1
.6..|..1...132...990..1848...990...132.....1
.7..|..1...182..2002..6006..6006..2002...182.....1
...

M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.

Cf. A002114, A086645, A186430, A186433 (inverse).

Table[2 Binomial[2 n, 2 k] - Boole[Or[k == 0, k == n]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 23 2017 *)

TABLE ENTRIES
T(n,k) = n!_S/(k!_S*(n-k)!_S),
which simplifies to
T(n,k) = 2*binomial(2*n,2*k) for 1 <= k < n,
with boundary conditions T(n,0) = 1 and T(n,n) = 1 for n >= 0.
RELATIONS WITH OTHER SEQUENCES
Denote this triangle by T. The first column of the inverse T^-1 (see A186433) begins [1, -1, 11, -301, 15371, ...] and, apart from the initial 1, is a signed version of the Glaisher's H' numbers A002114.
The first column of (1/2)*T^2 begins [1/2, 1, 7, 31, 127, ...] and, apart from the initial term, equals A000225(2*n-1), counting the preferential arrangements on (2*n - 1) labeled elements having less than or equal to two ranks.
The first column of (1/3)*T^3 begins [1/3, 1, 13, 181, 1933, ...] and, apart from the initial term, is A101052(2*n-1), which gives the number of preferential arrangements on (2*n-1) labeled elements having less than or equal to three ranks.

A371637 Triangle read by rows: T(n, k) = (-8)^kbinomial(2n, 2k)Euler(2*k, 1/2).

Original entry on oeis.org

1, 1, 2, 1, 12, 20, 1, 30, 300, 488, 1, 56, 1400, 13664, 22160, 1, 90, 4200, 102480, 997200, 1616672, 1, 132, 9900, 450912, 10969200, 106700352, 172976960, 1, 182, 20020, 1465464, 66546480, 1618288672, 15740903360, 25518205568

Offset: 0

Peter Luschny, Apr 02 2024

			Triangle starts:
[0] 1;
[1] 1,   2;
[2] 1,  12,    20;
[3] 1,  30,   300,     488;
[4] 1,  56,  1400,   13664,    22160;
[5] 1,  90,  4200,  102480,   997200,    1616672;
[6] 1, 132,  9900,  450912, 10969200,  106700352,   172976960;
[7] 1, 182, 20020, 1465464, 66546480, 1618288672, 15740903360, 25518205568;

Cf. A001105, A002939 (column 1), A012816 (main diagonal), A371683 (row sums), A371684 (alternating row sums).

Cf. A081658, A084938, A086645, A000079, A000364.

T := (n, k) -> (-8)^k*binomial(2*n, 2*k)*euler(2*k, 1/2):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

Table[(-8)^k*Binomial[2*n, 2*k]*EulerE[2*k, 1/2], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

def DelehamDelta(R, S, dim):
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [R(k) + x * S(k) for k in range(dim)]
    C = [ring(0)] + [ring(1) for i in range(dim)]
    for k in range(1, dim + 1):
        for n in range(k - 1, 0, -1):
            C[n] = C[n-1] + C[n+1] * A[n-1]
        yield list(C[1])
def A371637_triangle(dim):
    a = lambda n: 1 - n % 2
    b = lambda n: 2*(n + 1)^2
    for row in DelehamDelta(a, b, dim): print(row)
A371637_triangle(8)  # Peter Luschny, Apr 21 2024

Triangle T(n, k), 0 <= k <=n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [2, 8, 18, 32, 50, 72, 98, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 21 2024

T(n, k) = binomial(2*n, 2*k) * 2^k * abs(Euler(2*k)) = A086645(n, k) * A000079(k) * A000364(k). - Philippe Deléham, Apr 23 2024

A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1

Offset: 0

Gary W. Adamson, Apr 22 2008

ConvOffsStoT transform of the dodecahedral numbers A006566 starting (1, 20, 84, 220,...).
Row sums give A007613.
The matrix inverse starts:
  1;
  -1,1;
  19,-20,1;
  -1513,1596,-84,1;
  315523,-332860,17556,-220,1;
  -136085041,143562965,-7572565,95095,-455,1;
  105261234643,-111045393456,5857368972,-73562060,352716,-816,1; - R. J. Mathar, Mar 22 2013

			First few rows of the triangle are:
  [0] 1;
  [1] 1,   1;
  [2] 1,  20,     1;
  [3] 1,  84,    84,     1;
  [4] 1, 220,   924,   220,     1;
  [5] 1, 455,  5005,  5005,   455,   1;
  [6] 1, 816, 18564, 48620, 18564, 816, 1;
  ...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).

Cf. A006566, A007613, A086645, A034839, A070775, A177808.

Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)

More terms from Amiram Eldar, Jun 01 2025

cp's OEIS Frontend

A156740 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 7, read by rows.

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A156741 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 8, read by rows.

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A201461 Triangle read by rows: n-th row (n>=0) gives coefficients of the polynomial ((x+1)^(2^n) + (x-1)^(2^n))/2.

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A109446 Binomial coefficients C(n,k) with n-k even, read by rows.

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Maple

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A156739 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 6, read by rows.

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A156742 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 9, read by rows.

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A177808 Triangle T(n,m) = binomial(4*n, 4*m), 0 <= m <= n, read by rows.

A156740 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 7, read by rows.

A156741 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 8, read by rows.

A156739 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 6, read by rows.

A156742 Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2(n+j), 2(k+j))/binomial( 2(n-k+j), 2j) ), where m = 9, read by rows.

A177808 Triangle T(n,m) = binomial(4n, 4m), 0 <= m <= n, read by rows.

A371637 Triangle read by rows: T(n, k) = (-8)^kbinomial(2n, 2k)Euler(2*k, 1/2).

A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.