A056001 -id:A056001 - cp's OEIS frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Reinhard Zumkeller, Aug 30 2014

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A093565 (8,1) Pascal triangle.

Original entry on oeis.org

1, 8, 1, 8, 9, 1, 8, 17, 10, 1, 8, 25, 27, 11, 1, 8, 33, 52, 38, 12, 1, 8, 41, 85, 90, 50, 13, 1, 8, 49, 126, 175, 140, 63, 14, 1, 8, 57, 175, 301, 315, 203, 77, 15, 1, 8, 65, 232, 476, 616, 518, 280, 92, 16, 1, 8, 73, 297, 708, 1092, 1134, 798, 372, 108, 17, 1, 8, 81, 370, 1005

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(8;n,m) gives in the columns m>=1 the figurate numbers based on A017077, including the decagonal numbers A001107,(see the W. Lang link).
This is the eighth member, d=8, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-4, for d=1..7.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=(1+7*z)/(1-(1+x)*z).
The SW-NE diagonals give A022098(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 7. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins
  [1];
  [8,  1];
  [8,  9,  1];
  [8, 17, 10,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Lang, First 10 rows and array of figurate numbers .

Row sums: A005010(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 7 for n=2 and 0 else.
The column sequences give for m=1..9: A017077, A001107 (decagonal), A007585, A051797, A051878, A050404, A052226, A056001, A056122.
Cf. A093644 (d=9).

a093565 n k = a093565_tabl !! n !! k
a093565_row n = a093565_tabl !! n
a093565_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [8, 1]
-- Reinhard Zumkeller, Aug 31 2014

a(n, m)=F(8;n-m, m) for 0<= m <= n, otherwise 0, with F(8;0, 0)=1, F(8;n, 0)=8 if n>=1 and F(8;n, m):=(8*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=8 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+7*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 7*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(8 + 17*x + 10*x^2/2! + x^3/3!) = 8 + 25*x + 52*x^2/2! + 90*x^3/3! + 140*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A056122 a(n) = (8n+9)C(n+8,8)/9.

Original entry on oeis.org

1, 17, 125, 605, 2255, 7007, 19019, 46475, 104390, 218790, 432718, 814606, 1469650, 2554930, 4299130, 7027834, 11195503, 17425375, 26558675, 39714675, 58363305, 84412185, 120310125, 169169325, 234908700, 322420956, 437766252

Offset: 0

Barry E. Williams, Jul 06 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A056001.
Cf. A093565 ((8, 1) Pascal, column m=9). Partial sums of A056001.
Cf. similar sequences listed in A254142.

List([0..40], n-> (8*n+9)*Binomial(n+8,8)/9); # G. C. Greubel, Aug 29 2019

[(8*n+9)*Binomial(n+8,8)/9: n in [0..40]]; // G. C. Greubel, Aug 29 2019

seq((8*n+9)*binomial(n+8,8)/9, n=0..40); # G. C. Greubel, Aug 29 2019

Table[(8n+9) Binomial[n+8,8]/9,{n,0,40}]  (* Harvey P. Dale, Mar 09 2011 *)

vector(40, n, (8*n+1)*binomial(n+7,8)/9) \\ G. C. Greubel, Aug 29 2019

[(8*n+9)*binomial(n+8,8)/9 for n in (0..40)] # G. C. Greubel, Aug 29 2019

G.f.: (1+7*x)/(1-x)^10.
a(n) = (362880 + 1308816*n + 1939788*n^2 + 1550548*n^3 + 740313*n^4 + 220416*n^5 + 41202*n^6 + 4692*n^7 + 297*n^8 + 8*n^9)/362880. - Harvey P. Dale, Mar 09 2011
E.g.f.: (362880 +5806080*x +16692480*x^2 +16934400*x^3 +7832160*x^4 + 1862784*x^5 +239904 x^6 +16704*x^7 +585*x^8 +8*x^9)*exp(x)/362880. - G. C. Greubel, Aug 29 2019

cp's OEIS Frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

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A093565 (8,1) Pascal triangle.

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A056122 a(n) = (8n+9)C(n+8,8)/9.

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cp's OEIS Frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

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A093565 (8,1) Pascal triangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Formula

A056122 a(n) = (8*n+9)*C(n+8,8)/9.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A056122 a(n) = (8n+9)C(n+8,8)/9.