cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A357339 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268437(n - k, j).

Original entry on oeis.org

1, -1, 1, 10, -2, 1, -270, 24, -3, 1, 14056, -720, 44, -4, 1, -1197000, 40320, -1500, 70, -5, 1, 151169040, -3628800, 92064, -2700, 102, -6, 1, -26521775280, 479001600, -8890560, 181888, -4410, 140, -7, 1, 6169461217920, -87178291200, 1241982720, -18910080, 324912, -6720, 184, -8, 1
Offset: 0

Views

Author

Peter Luschny, Sep 25 2022

Keywords

Examples

			Triangle starts:
[0]         1;
[1]        -1,        1;
[2]        10,       -2,     1;
[3]      -270,       24,    -3,     1;
[4]     14056,     -720,    44,    -4,   1;
[5]  -1197000,    40320, -1500,    70,  -5,  1;
[6] 151169040, -3628800, 92064, -2700, 102, -6, 1;

Crossrefs

Cf. A357342 (alternating row sums), A268437, A357340.

Programs

  • Maple
    A357339 := proc(n, k) local u; u:=(n - k); (2*u)!*add(binomial(-n, j) * j! * add((-1)^(j+m)*binomial(u+j, u+m)*Stirling2(u+m, m), m=0..j) / (u+j)!, j=0..u) end: seq(print(seq(A357339(n, k), k=0..n)), n=0..6);
  • SageMath
    # using function A268437.
def A357339(n, k):
    return sum(binomial(-n, i) * A268437(n - k, i) for i in range(n - k + 1))
for n in range(9): print([A357339(n, k) for k in range(n + 1)])

A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025
Offset: 0

Views

Author

Peter Luschny, Mar 26 2016

Keywords

Comments

We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022

Examples

			Triangle starts:
  1;
  0, 1;
  0, 1,   3;
  0, 1,  10,   15;
  0, 1,  25,  105,   105;
  0, 1,  56,  490,  1260,    945;
  0, 1, 119, 1918,  9450,  17325,  10395;
  0, 1, 246, 6825, 56980, 190575, 270270, 135135;

Links

Crossrefs

Variants: A134991 (main entry for this triangle), A181996.
Row sums are A000311.
Alternating row sums are signed factorials A133942.
Cf. A269940 (Stirling1 counterpart), A268437.

Programs

  • Maple
    # first version
A269939 := (n,k) -> add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m, m), m=0..k):
seq(seq(A269939(n,k), k=0..n), n=0..8);
# Alternatively:
T := proc(n,k) option remember;
    `if`(k=0 and n=0, 1,
    `if`(k<=0 or k>n, 0,
    k*T(n-1,k)+(n+k-1)*T(n-1,k-1))) end:
for n from 0 to 6 do seq(T(n,k),k=0..n) od;
# simple, third version
T := (n,k)->  (n+k)!*coeftayl((exp(z)-z-1)^k/k!, z=0, n+k); # Marko Riedel, Apr 14 2016
  • Mathematica
    Table[Sum[(-1)^(m + k) Binomial[n + k, n + m] StirlingS2[n + m, m], {m, 0, k}], {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 15 2016 *)
  • PARI
    T(n) = {[Vecrev(Pol(p)) | p<-Vec(serlaplace(1/((1+y)*(1 + lambertw(-y/(1+y)*exp((x-y)/(1+y) + O(x*x^n)))))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022
  • Sage
    T = lambda n,k: sum((-1)^(m+k)*binomial(n+k,n+m)*stirling_number2(n+m,m) for m in (0..k))
for n in (0..6): print([T(n,k) for k in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

Formula

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](1/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.
T(n,k) = A268437(n,k)*FF(n+k,n)/(2*n)!.
T(n,k) = (n+k)! [z^{n+k}] (exp(z)-z-1)^k/k!. - Marko Riedel, Apr 14 2016
From Fabián Pereyra, Jan 12 2022: (Start)
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) for n > 0, T(0,0) = 1, T(n,0) = 0 for n > 0. (See the second Maple program.)
E.g.f.: A(x,t) = 1/((1+t)*(1 + W(-t/(1+t)*exp((x-t)/(1+t))))), where W(x) is the Lambert W-function.
T(n,k) = Sum_{j=0..k} E2(n,j)*binomial(n-j,k-j), where E2(n,k) are the second-order Eulerian numbers A340556.
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A112486(n,j)*binomial(j,k). (End)

A268438 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 6, 0, 180, 240, 90, 0, 8064, 14560, 10080, 2520, 0, 604800, 1330560, 1285200, 604800, 113400, 0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400, 0, 10897286400, 30858347520, 43770767040, 36970053120, 18918900000, 5448643200, 681080400
Offset: 0

Views

Author

Peter Luschny, Mar 07 2016

Keywords

Comments

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

Examples

			Triangle starts:
[1],
[0, 1],
[0, 8,        6],
[0, 180,      240,       90],
[0, 8064,     14560,     10080,     2520],
[0, 604800,   1330560,   1285200,   604800,    113400],
[0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400].

Links

Crossrefs

Cf. A000680, A060593, A268437, A269940, A269941.

Programs

  • Maple
    A268438 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n, k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(n+k-1)*(T(n-1,k)+T(n-1,k-1)))) end:
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268438(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268438_row := n -> PTrans(n, n->n/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(lprint(A268438_row(n)), n=0..8);
  • Mathematica
    T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)
  • Sage
    A268438 = lambda n,k: (factorial(2*n)/falling_factorial(n+k,n))*sum((-1)^(m+k)* binomial(n+k,n+m)*stirling_number1(n+m,m) for m in (0..k))
for n in (0..7): print([A268438(n,m) for m in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n))

Formula

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269940(n,k).
T(n,1) = (2*n)!/(n+1) = A060593(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A268435 Triangle read by rows, T(n,k) = RF(n-k+1,n-k)*S2(n,k) where RF denotes the rising factorial and S2 the Stirling set numbers, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 12, 6, 1, 0, 120, 84, 12, 1, 0, 1680, 1800, 300, 20, 1, 0, 30240, 52080, 10800, 780, 30, 1, 0, 665280, 1905120, 505680, 42000, 1680, 42, 1, 0, 17297280, 84490560, 29211840, 2857680, 126000, 3192, 56, 1
Offset: 0

Views

Author

Peter Luschny, Mar 07 2016

Keywords

Examples

			[1]
[0,      1]
[0,      2,       1]
[0,     12,       6,      1]
[0,    120,      84,     12,     1]
[0,   1680,    1800,    300,    20,    1]
[0,  30240,   52080,  10800,   780,   30,  1]
[0, 665280, 1905120, 505680, 42000, 1680, 42, 1]

Links

Crossrefs

Cf. A048993, A268436, A268437, A268438.

Programs

  • Maple
    T := (n, k) -> pochhammer(n-k+1,n-k)*Stirling2(n,k):
for n from 0 to 9 do seq(T(n,k), k=0..n) od;
# Alternatively:
T := proc(n,k) option remember;
  `if`( n=k, 1,
  `if`( k=0, 0,
   k*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1))) end:
for n from 0 to 7 do seq(T(n,k),k=0..n) od;
  • Mathematica
    T[n_, k_] := Pochhammer[n-k+1, n-k] StirlingS2[n, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
  • Sage
    A268435 = lambda n,k: rising_factorial(n-k+1,n-k)*stirling_number2(n,k)
[[A268435(n,k) for k in (0..n)] for n in range(8)]

Formula

T(n,k) = binomial(n,k)*Sum_{i=0..k} binomial(k,i)*A268437(n-k,i).
T(n,k) = binomial(-k,-n)*Sum_{i=0..n-k} binomial(-n,i)*A268438(n-k,i).
T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A048993(n,k)/sqrt(Pi).
T(n,1) = (2*n-2)!/(n-1)! for n>=1.
T(n,n-1) = (n-1)*n for n>=1.
Recurrence: T(n,k) = 1 if k=n; 0 if k=0; otherwise k*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1).

A268436 Triangle read by rows, T(n,k) = RF(n-k+1, n-k) * S1(n,k) where RF denotes the rising factorial and S1 the Stirling cycle numbers, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 24, 6, 1, 0, 720, 132, 12, 1, 0, 40320, 6000, 420, 20, 1, 0, 3628800, 460320, 27000, 1020, 30, 1, 0, 479001600, 53343360, 2728320, 88200, 2100, 42, 1, 0, 87178291200, 8693879040, 397111680, 11371920, 235200, 3864, 56, 1
Offset: 0

Views

Author

Peter Luschny, Mar 07 2016

Keywords

Examples

			[1]
[0,         1]
[0,         2,        1]
[0,        24,        6,       1]
[0,       720,      132,      12,     1]
[0,     40320,     6000,     420,    20,    1]
[0,   3628800,   460320,   27000,  1020,   30,  1]
[0, 479001600, 53343360, 2728320, 88200, 2100, 42, 1]

Links

Crossrefs

Cf. A132393, A268435, A268437, A268438.

Programs

  • Maple
    T := (n,k) -> pochhammer(n-k+1, n-k)*abs(Stirling1(n,k)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od;
# Alternatively:
T := proc(n,k) option remember;
  `if`( n=k, 1,
  `if`( k=0, 0,
   (n-1)*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1))) end:
for n from 0 to 7 do seq(T(n,k),k=0..n) od;
  • Mathematica
    T[n_, k_] := Pochhammer[n-k+1, n-k] Abs[StirlingS1[n, k]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)
  • Sage
    A268436 = lambda n,k: rising_factorial(n-k+1,n-k)*stirling_number1(n,k)
[[A268436(n,k) for k in (0..n)] for n in range(8)]

Formula

T(n,k) = binomial(n,k)*Sum_{i=0..k} binomial(k,i)*A268438(n-k,i).
T(n,k) = binomial(-k,-n)*Sum_{i=0..n-k} binomial(-n,i)*A268437(n-k,i).
T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A132393(n,k)/sqrt(Pi).
T(n,1) = (2*n-2)! for n>=1.
T(n,n-1) = (n-1)*n for n>=1.
Recurrence: T(n,k) = 1 if k=n; 0 if k=0; and otherwise (n-1)*(4*(n-k)-2)*T(n-1,k) + T(n-1,k-1).

A268439 Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling2(n+m,m), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 4, 3, 0, 15, 60, 15, 0, 56, 700, 840, 105, 0, 210, 6720, 22050, 12600, 945, 0, 792, 58905, 421960, 623700, 207900, 10395, 0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135, 0, 11440, 4012008, 100180080, 551450900, 916515600, 491891400, 75675600, 2027025
Offset: 0

Views

Author

Peter Luschny, Mar 08 2016

Keywords

Examples

			[1]
[0, 1]
[0, 4, 3]
[0, 15, 60, 15]
[0, 56, 700, 840, 105]
[0, 210, 6720, 22050, 12600, 945]
[0, 792, 58905, 421960, 623700, 207900, 10395]
[0, 3003, 492492, 6831825, 20740720, 17342325, 3783780, 135135]

Links

Crossrefs

Cf. A001147, A001791, A268437, A268440, A269939, A269941.

Programs

  • Maple
    # The function PTrans is defined in A269941.
A268439_row := n -> PTrans(n, n->1/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
seq(print(A268439_row(n)), n=0..8);
  • Sage
    A268439 = lambda n, k: binomial(2*n, n+k)*sum((-1)^(m+k)*binomial(n+k, n+m)* stirling_number2(n+m, m) for m in (0..k))
for n in (0..7): print([A268439(n, m) for m in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
# Alternatively
PtransMatrix(8, lambda n: 1/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))

Formula

T(n,k) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](1/(n+1)) where P is the P-transform. The P-transform is defined in the link.
T(n,k) = A269939(n,k)*binomial(2*n,n+k).
T(n,k) = A268437(n,k)/(k!*(n-k)!).
T(n,1) = binomial(2*n,n-1) = A001791(n) for n>=1.
T(n,n) = (2*n-1)!! = A001147(n) for n>=0.
Showing 1-6 of 6 results.

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