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.

A357340 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268438(n - k, j).

Original entry on oeis.org

1, -1, 1, 2, -2, 1, 0, 12, -3, 1, -56, -120, 28, -4, 1, 0, 1680, -450, 50, -5, 1, 15840, -30240, 10416, -1080, 78, -6, 1, 0, 665280, -317520, 33712, -2100, 112, -7, 1, -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1
Offset: 0

Views

Author

Peter Luschny, Sep 25 2022

Keywords

Examples

			Triangle T(n, k) starts:
[0]         1;
[1]        -1,         1;
[2]         2,        -2,        1;
[3]         0,        12,       -3,        1;
[4]       -56,      -120,       28,       -4,     1;
[5]         0,      1680,     -450,       50,    -5,     1;
[6]     15840,    -30240,    10416,    -1080,    78,    -6,   1;
[7]         0,    665280,  -317520,    33712, -2100,   112,  -7,  1;
[8] -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1;

Crossrefs

Cf. A357341 (alternating row sums), A264437, A268438, A357339.

Programs

  • Maple
    A357340 := proc(n, k) local u; u := n - k; (2*u)!*add(binomial(-n, j) * j! *
  add((-1)^(j+m)*binomial(u+j, u+m)*abs(Stirling1(u+m, m)), m=0..j)/(u +j)!, j=0..u) end: seq(print(seq(A357340(n, k), k=0..n)), n=0..8);
  • SageMath
    # using function A268438
def A357340(n, k):
    return sum(binomial(-n, i) * A268438(n - k, i) for i in range(n - k + 1))
for n in range(10): print([A357340(n, k) for k in range(n + 1)])

A268437 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](1/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 4, 6, 0, 30, 120, 90, 0, 336, 2800, 5040, 2520, 0, 5040, 80640, 264600, 302400, 113400, 0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400, 0, 2162160, 118198080, 983782800, 2986663680, 4162158000, 2724321600, 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

			[1],
[0, 1],
[0, 4, 6],
[0, 30, 120, 90],
[0, 336, 2800, 5040, 2520],
[0, 5040, 80640, 264600, 302400, 113400],
[0, 95040, 2827440, 15190560, 29937600, 24948000, 7484400].

Links

Crossrefs

Cf. A000680, A001761, A268438, A269939, A269941.

Programs

  • Maple
    A268437 := 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*(k*T(n-1,k)+(n+k-1)*T(n-1,k-1)))) end;
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268437(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268437_row := n -> PTrans(n, n->1/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(print(A268437_row(n)),n=0..8);
  • Mathematica
    T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] StirlingS2[n+m, m], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)
  • Sage
    A268437 = lambda n, k: (factorial(2*n)/falling_factorial(n+k, n))*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([A268437(n, m) for m in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(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)*Stirling2(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269939(n,k).
T(n,1) = (2*n)!/(n+1)! = A001761(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)*binomial(n + k, n + m) * |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 6, 20, 15, 0, 24, 130, 210, 105, 0, 120, 924, 2380, 2520, 945, 0, 720, 7308, 26432, 44100, 34650, 10395, 0, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025
Offset: 0

Views

Author

Peter Luschny, Mar 27 2016

Keywords

Comments

We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022

Examples

			Triangle T(n,k) starts:
  [1]
  [0,   1]
  [0,   2,      3]
  [0,   6,     20,     15]
  [0,  24,    130,    210,    105]
  [0,  120,   924,   2380,   2520,    945]
  [0,  720,  7308,  26432,  44100,  34650,  10395]
  [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135]

Links

Crossrefs

Variants: A111999, A259456.
Cf. A269939 (Stirling2 counterpart), A268438, A032188 (row sums).

Programs

  • Maple
    T := (n, k) -> add((-1)^(m+k)*binomial(n+k,n+m)*abs(Stirling1(n+m, m)), m=0..k):
seq(print(seq(T(n, k), k=0..n)), n=0..6);
# Alternatively:
T := proc(n, k) option remember;
    `if`(k=0, k^n,
    `if`(k<=0 or k>n, 0,
    (n+k-1)*(T(n-1, k)+T(n-1, k-1)))) end:
for n from 0 to 6 do seq(T(n, k), k=0..n) od;
  • Mathematica
    T[n_, k_] := Sum[(-1)^(m+k)*Binomial[n+k, n+m]*Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 12 2022 *)
  • Sage
    T = lambda n, k: 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([T(n, k) for k in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: n/(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](n/(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) = A268438(n,k)*FF(n+k,n)/(2*n)!.

Extensions

Name corrected after notice from Ed Veling by Peter Luschny, Jun 14 2022

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).

A268440 Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling1(n+m,m), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 3, 0, 90, 120, 15, 0, 1344, 3640, 1680, 105, 0, 25200, 110880, 107100, 25200, 945, 0, 570240, 3617460, 5815040, 2910600, 415800, 10395, 0, 15135120, 128576448, 303963660, 256736480, 78828750, 7567560, 135135
Offset: 0

Views

Author

Peter Luschny, Mar 08 2016

Keywords

Examples

			[1]
[0, 1]
[0, 8, 3]
[0, 90, 120, 15]
[0, 1344, 3640, 1680, 105]
[0, 25200, 110880, 107100, 25200, 945]
[0, 570240, 3617460, 5815040, 2910600, 415800, 10395]

Links

Crossrefs

Cf. A001147, A092956, A268438, A268439, A269940, A269941.

Programs

  • Maple
    # The function PTrans is defined in A269941.
A268440_row := n -> PTrans(n, n->n/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
seq(print(A268440_row(n)), n=0..8);
  • Sage
    A268440 = lambda n, k: binomial(2*n,n+k)*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([A268440(n, m) for m in (0..n)])
  • Sage
    # uses[PtransMatrix from A269941]
# Alternatively
PtransMatrix(7, lambda n: n/(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](n/(n+1)) where P is the P-transform. The P-transform is defined in the link.
T(n,k) = A269940*binomial(2*n,n+k).
T(n,k) = A268438(n,k)/(k!*(n-k)!).
T(n,1) = n*(2*n)!/(n+1)! for n>=1 (cf. A092956).
T(n,n) = (2*n-1)!! = A001147(n) for n>=0.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.