Harlan J. Brothers - cp's OEIS frontend

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

Offset: 0

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].
For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.
The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.
The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).
The row sums give n! = A000142(n).

			The triangle T begins:
n\k    0       1        2         3         4          5          6         7
0:     1
1:    -3       4
2:     9     -32       25
3:   -27     192     -375       216
4:    81   -1024     3750     -5184      2401
5:  -243    5120   -31250     77760    -84035      32768
6:   729  -24576   234375   -933120   1764735   -1572864     531441
7: -2187  114688 -1640625   9797760 -28824005   44040192  -33480783  10000000
...
n = 8:  6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,
n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)
Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n,k](k+3)^n,{k,0,n}];Array[A362353row,10,0] (* Paolo Xausa, Jul 30 2023 *)

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.
O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.
E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.
E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

a(41)-a(44) from Paolo Xausa, Jul 31 2023

A191510 Product of terms in n-th row of A132818.

Original entry on oeis.org

1, 9, 648, 360000, 1518750000, 48243443062500, 11480517255997440000, 20400479323264014247526400, 270090559531318533654528000000000, 26599911685677709861296622500000000000000, 19464564507161243794359748945629699456000000000000

Offset: 1

Harlan J. Brothers, Jun 04 2011

Lim_{n -> inf} (a(n)*a(n+2))/a(n+1)^2 = e^2. Like A168510, this limit is asymptotic from above.

			For n=3, row 3 of A132818 = {6,18,6} and a(3)=648.

H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, Vol. 20, No. 4, (1998), 25-29.

Cf. A132818, A002457. Related to e as in the cases of A168510 and A001142.

Table[Product[Product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}], {n, 1, 11}]
Table[(n + 1)^n * Hyperfactorial[n]^2 / (2^n * BarnesG[n+2]^2), {n, 1, 12}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n)=product[product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}].

a(n) ~ A^4 * exp(n^2 + 2*n + 5/6) / (n^(2/3) * 2^(2*n+1) * Pi^(n+1)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1

Offset: 0

Harlan J. Brothers, Jun 16 2010

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P for n = {1, 2, 3, ..., k}.

			Prism begins (levels 1-4):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Level 1 = A007318.
Level 2 = A003506.
Level 3 = A094305.
Level 4 = A178820.
Level 5 = A178821.
Level 6 = A178822.
Sums of shallow diagonals for each level correspond to rows of square A037027.
Contains A109649 and A046816.
P = A000984.
P = A006480.
P = A000897.
P<3n-2, 3n-2, n> = A113424.

end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).

Keyword tabf by Michel Marcus, Oct 22 2017

A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 4, 10, 20, 10, 20, 60, 60, 20, 35, 140, 210, 140, 35, 56, 280, 560, 560, 280, 56, 84, 504, 1260, 1680, 1260, 504, 84, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220

Offset: 0

Harlan J. Brothers, Jun 17 2010

The product of the tetrahedral numbers (A000292, beginning with second term) and Pascal's triangle (A007318). Also level 4 of Pascal's prism (A178819): (i+3; 3, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
   1;
   4,   4;
  10,  20,  10;
  20,  60,  60,  20;
  35, 140, 210, 140,  35;

G. C. Greubel, Rows n=0..100 of triangle, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000292, A007318, A178819.

Rows sums give A001789.

T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+3, 3)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

/* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017

T:=(n,k)->binomial(n+3,3)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019

Table[Multinomial[3, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

{T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019

[[binomial(n+3, 3)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(4, i, j) = (i+2; 3, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^4. - Ilya Gutkovskiy, Mar 20 2020

A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 5, 5, 15, 30, 15, 35, 105, 105, 35, 70, 280, 420, 280, 70, 126, 630, 1260, 1260, 630, 126, 210, 1260, 3150, 4200, 3150, 1260, 210, 330, 2310, 6930, 11550, 11550, 6930, 2310, 330, 495, 3960, 13860, 27720, 34650, 27720, 13860, 3960, 495, 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715

Offset: 0

Harlan J. Brothers, Jun 19 2010

The product of the pentatope numbers (A000332, beginning with fifth term) and Pascal's triangle (A007318). Also level 5 of Pascal's prism (A178819) read by rows: (i+4; 4, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
   1;
   5,   5;
  15,  30,  15;
  35, 105, 105,  35;
  70, 280, 420, 280,  70;

G. C. Greubel, Rows n=0..100 of triangle, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000332, A007318, A178819, A178820, A178822.

Rows sum to A003472, shallow diagonals sum to A001873.

T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+4, 4)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

/* As triangle */ [[Binomial(n+4,4)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017

T:=(n,k)->binomial(n+4,4)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019

Table[Multinomial[4, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

{T(n,k) = binomial(n+4, 4)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019

[[binomial(n+4, 4)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

T(n,k) = C(n+4,4) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(5, i, j) = (i+3; 4, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^5. - Ilya Gutkovskiy, Mar 20 2020

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 6, 6, 21, 42, 21, 56, 168, 168, 56, 126, 504, 756, 504, 126, 252, 1260, 2520, 2520, 1260, 252, 462, 2772, 6930, 9240, 6930, 2772, 462, 792, 5544, 16632, 27720, 27720, 16632, 5544, 792, 1287, 10296, 36036, 72072, 90090, 72072, 36036, 10296, 1287

Offset: 0

Harlan J. Brothers, Jun 19 2010

The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
    1;
    6,   6;
   21,  42,  21;
   56, 168, 168,  56;
  126, 504, 756, 504, 126;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000389, A007318, A178819, A178820, A178821.

Rows sum to A054849, shallow diagonals sum to A001874.

/* As triangle */ [[Binomial(n+5,5)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Oct 23 2017

Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
Table[Binomial[n + 5, 5]*Binomial[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

for(n=0,10, for(k=0,n, print1(binomial(n+5,5)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 25 2017

T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^6. - Ilya Gutkovskiy, Mar 20 2020

A168510 Products across consecutive rows of the denominators of the Leibniz harmonic triangle (A003506).

Original entry on oeis.org

1, 4, 54, 2304, 300000, 116640000, 133413966000, 444110104166400, 4267295479315169280, 117595223746560000000000, 9245836018244425723200000000, 2065215715357207851951980544000000

Offset: 1

Harlan J. Brothers, Nov 27 2009

As in A001142, lim_{n->inf} (a(n)a(n+2))/a(n+1)^2 = e, demonstrating an underlying relation between A003506 and Pascal's triangle A007318. Unlike A001142, in this case the function is asymptotic from above.

			For n=3, row 3 of A003506 = {3, 6, 3} and a(3)=54.
a(5) = 5^5 * 4^3 * 3^1 * 2^-1 * 1^-3 = 5^5 * 3 * 2^5 = 300000. - _Peter Munn_, Mar 07 2018

A. Bogomolny, Cut The Knot: Leibniz and Pascal Triangles
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.

Cf. A003506, A001142, A007318. For n >= 1, a(n) = n!*A001142(n).

Table[n! Product[k^(2 k - n - 1), {k, 1, n}], {n, 1, 12}]
Table[Product[Product[(1 - 1/k)^-k, {k, 2, j}], {j, 1, n}], {n, 1, 12}]
(* or *)
a[1] = 1; a[n_] := a[n - 1] Product[(1 - 1/k)^-k, {k, 2, n}]

a(n) = n!*Product_{k=1..n} k^(2k-n-1).
a(n) = Product_{j=1..n} Product_{k=2..j} ((1-1/k)^-k).
a(1) = 1; a(n) = a(n-1)*Product_{k=2..n} ((1-1/k)^-k).
a(n) ~ A^2 * exp(n^2/2 - 1/12) * n^(n/2 + 1/6) / (2*Pi)^(n/2), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 22 2017
a(n) = Product_{k=0..n-1} (n-k)^(n-2k). - Peter Munn, Mar 07 2018

A141052 Number of runs or rising sequences of length 2 among all permutations of n.

Original entry on oeis.org

1, 4, 21, 130, 930, 7560, 68880, 695520, 7711200, 93139200, 1217462400, 17124307200, 257902444800, 4140968832000, 70614415872000, 1274546617344000, 24275666967552000, 486580401635328000, 10238462617743360000, 225651661258383360000, 5198503365971435520000

Offset: 2

Harlan J. Brothers, Jul 31 2008, Aug 24 2008

			a[3]=4 because of the 6 permutations of n=3, there are 4 ascending runs of length 2:
{1,3} in {1,3,2}
{1,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}
a[3]=4 because of the 6 permutations of n=3, there are 4 rising sequences of length 2:
{1,2} in {1,3,2}
{2,3} in {2,1,3}
{2,3} in {2,3,1}
{1,2} in {3,1,2}

Persi Diaconis, Mathematical developments from the analysis of riffle shuffling, p. 4.
Francis Edward Su, Rising Sequences in Card Shuffling
Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, American Mathematical Society, 1997, pp.120-131.

Column 2 of A122843.

Cf. A008292, A097900, A001286, A001048, A000142, A028387, A001710.

Table[n!(5n + 1)/4! + Floor[2/n](1/12), {n, 2, 10}]

a(n) = n!*(5n+1)/4! + floor(2/n)*(1/12), n>=2.
Recurrence: a(n) = (n+1)*a(n-1)+(n-1)!/6, n>=2, with a(2)=1 and a(3)=4.
E.g.f.: x^2*(x-2)*(x-6)/(24*(x-1)^2).

First example and typo in second example corrected by Harlan J. Brothers, Apr 29 2013

cp's OEIS Frontend

User: Harlan J. Brothers

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)*binomial(n, k)*(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A191510 Product of terms in n-th row of A132818.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A362353 Triangle read by rows: T(n,k) = (-1)^(n-k)binomial(n, k)(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.