A035277 -id:A035277 - cp's OEIS frontend

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A045757 10-factorial numbers.

Original entry on oeis.org

1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..323
Index entries for sequences related to factorial numbers.

Cf. A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

Cf. A008542, A048176.

List([1..20], n-> Product([0..n-1], j-> 10*j+1) ); # G. C. Greubel, Nov 11 2019

[(&*[10*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

G(x):=-1+(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..14); # Zerinvary Lajos, Apr 03 2009
seq(mul(10*j+1, j = 0..n-1), n = 1..20); # G. C. Greubel, Nov 11 2019

FoldList[Times,10*Range[0,20]+1] (* Harvey P. Dale, Dec 02 2016 *)

vector(21, n, prod(j=0,n-1, 10*j+1) ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j+1) for j in (0..n-1)) for n in (1..20)] # G. C. Greubel, Nov 11 2019

a(n) = Pochhammer(1/10,n)*10^n.
a(n+1) = (10*n+1)(!^10) = Product_{k=0..n} (10*k+1), n >= 0.
E.g.f.: -1 + (1-10*x)^(-1/10).
Sum_{n>=1} 1/a(n) = (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035278 One ninth of deca-factorial numbers.

Original entry on oeis.org

1, 19, 551, 21489, 1052961, 62124699, 4286604231, 338641734249, 30139114348161, 2983772320467939, 325231182931005351, 38702510768789636769, 4992623889173863143201, 693974720595166976904939, 103402233368679879558835911, 16440955105620100849854909849

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A045757, A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

List([1..20], n-> Product([1..n], j-> 10*j-1)/9 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-1: j in [1..n]])/9: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-1, j=1..n)/9, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[9/10, n]/9, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-1)/9 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-1) for j in (1..n))/9 for n in (1..20)] # G. C. Greubel, Nov 11 2019

9*a(n) = (10*n-1)(!^10) = Product_{j=1..n} (10*j-1).
E.g.f.: (-1+(1-10*x)^(-9/10))/9.
a(n) = (Pochhammer(9/10,n) * 10^n)/9.
Sum_{n>=1} 1/a(n) = 9*(e/10)^(1/10)*(Gamma(9/10) - Gamma(9/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035273 One quarter of deca-factorial numbers.

Original entry on oeis.org

1, 14, 336, 11424, 502656, 27143424, 1737179136, 128551256064, 10798305509376, 1015040717881344, 105564234659659776, 12034322751201214464, 1492256021148950593536, 199962306833959379533824, 28794572184090150652870656, 4434364116349883200542081024

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A034323, A035272, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-6)/4 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-6: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-6, j=1..n)/4, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[4/10, n]/4, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-6)/4 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-6) for j in (1..n))/4 for n in (1..20)] # G. C. Greubel, Nov 11 2019

4*a(n) = (10*n-6)(!^10) = Product_{j=1..n} (10*j-6).
a(n) = 2^(n+1)*A034323(n) where 2*A034323(n)= (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
E.g.f.: (-1 + (1-10*x)^(-2/5))/4.
a(n) = (Pochhammer(4/10,n)*10^n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/10^6)^(1/10)*(Gamma(2/5) - Gamma(2/5, 1/10)). - Amiram Eldar, Dec 22 2022

A035272 One third of deca-factorial numbers.

Original entry on oeis.org

1, 13, 299, 9867, 424281, 22486893, 1416674259, 103417220907, 8583629335281, 798277528181133, 82222585402656699, 9291152150500206987, 1142811714511525459401, 151993958030032886100333, 21735135998294702712347619, 3325475807739089514989185707, 542052556661471590943237270241

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-7)/3 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-7: j in [1..n]])/3: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-7, j=1..n)/3, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[3/10, n]/3, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-7)/3 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-7) for j in (1..n))/3 for n in (1..20)] # G. C. Greubel, Nov 11 2019

3*a(n) = (10*n-7)(!^10) = Product_{j=1..n} (10*j-7).
E.g.f.: (-1 + (1-10*x)^(-3/10))/3.
Sum_{n>=1} 1/a(n) = 3*(e/10^7)^(1/10)*(Gamma(3/10) - Gamma(3/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035274 One fifth of deca-factorial numbers.

Original entry on oeis.org

1, 15, 375, 13125, 590625, 32484375, 2111484375, 158361328125, 13460712890625, 1278767724609375, 134270611083984375, 15441120274658203125, 1930140034332275390625, 260568904634857177734375, 37782491172054290771484375, 5856286131668415069580078125

Offset: 1

Wolfdieter Lang

a(n)= (Pochhammer(5/10,n)*10^n)/5.

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A001147, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-5)/5 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-5: j in [1..n]])/5: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-5, j=1..n)/5, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[5/10, n]/5, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-5)/5 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-5) for j in (1..n))/5 for n in (1..20)] # G. C. Greubel, Nov 11 2019

5*a(n) = (10*n-5)(!^10) = Product_{j=1..n} (10*j-5).
a(n) = 5^n*A001147(n) where A001147(n) = (2*n-1)!!.
E.g.f.: (-1 + (1-10*x)^(-1/2))/5.
a(n) = (Pochhammer(5/10,n)*10^n)/5.
Sum_{n>=1} 1/a(n) = exp(1/10)*sqrt(5*Pi/2)*erf(1/sqrt(10)), where erf is the error function. - Amiram Eldar, Dec 22 2022

A035275 One sixth of deca-factorial numbers.

Original entry on oeis.org

1, 16, 416, 14976, 688896, 38578176, 2546159616, 193508130816, 16641699250176, 1597603128016896, 169345931569790976, 19644128062095753216, 2475160135824064905216, 336621778472072827109376, 49146779656922632757968896, 7666897626479930710243147776

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A034300, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-4)/6 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-4: j in [1..n]])/6: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-4, j=1..n)/6, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[6/10, n]/6, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-4)/6 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-4) for j in (1..n))/6 for n in (1..20)] # G. C. Greubel, Nov 11 2019

6*a(n) = (10*n-4)(!^10) = Product_{j=1..n} (10*j-4).
a(n) = 2^n*3*A034300(n) where 3*A034300(n) = (5*n-2)(!^5).
E.g.f.: (-1 + (1-10*x)^(-3/5))/6.
a(n) = (Pochhammer(6/10,n) * 10^n)/6.
Sum_{n>=1} 1/a(n) = 6*(e/10^4)^(1/10)*(Gamma(3/5) - Gamma(3/5, 1/10)). - Amiram Eldar, Dec 22 2022

A035276 One seventh of deca-factorial numbers.

Original entry on oeis.org

1, 17, 459, 16983, 798201, 45497457, 3048329619, 234721380663, 20420760117681, 1980813731415057, 211947069261411099, 24797807103585098583, 3149321502155307520041, 431457045795277130245617, 63424185731905738146105699, 9957597159909200888938594743

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-3)/7 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-3: j in [1..n]])/7: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-3, j=1..n)/7, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[7/10, n]/7, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-3)/7 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-3) for j in (1..n))/7 for n in (1..20)] # G. C. Greubel, Nov 11 2019

7*a(n) = (10*n-3)(!^10) = Product_{j=1..n} (10*j-3).
E.g.f.: (-1 + (1-10*x)^(-7/10))/7.
a(n) = (Pochhammer(7/10,n)*10^n)/7.
Sum_{n>=1} 1/a(n) = 7*(e/10^3)^(1/10)*(Gamma(7/10) - Gamma(7/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035265 One half of deca-factorial numbers.

Original entry on oeis.org

1, 12, 264, 8448, 354816, 18450432, 1143926784, 82362728448, 6753743732736, 621344423411712, 63377131187994624, 7098238693055397888, 865985120552758542336, 114310035912964127588352, 16232025099640906117545984, 2467267815145417729866989568, 399697386053557672238452310016

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A008548, A045757, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

List([1..20], n-> Product([1..n], j-> 10*j-8)/2 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-8: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-8, j=1..n)/2, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[2/10, n]/2, {n,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-8)/2 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-8) for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Nov 11 2019

2*a(n) = (10*n-8)(!^10) = Product_{j=1..n} (10*j-8).
a(n) = 2^n*A008548(n) where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).
E.g.f.: (-1 + (1-10*x)^(-1/5))/2.
a(n) = (Pochhammer(2/10,n)*10^n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/10^8)^(1/10)*(Gamma(1/5) - Gamma(1/5, 1/10)). - Amiram Eldar, Dec 22 2022

cp's OEIS Frontend

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A045757 10-factorial numbers.

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A035278 One ninth of deca-factorial numbers.

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A035273 One quarter of deca-factorial numbers.

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A035272 One third of deca-factorial numbers.

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A035274 One fifth of deca-factorial numbers.

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.