A002452 -id:A002452 - cp's OEIS frontend

A218736 a(n) = (33^n - 1)/32.

0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 33 (A009977).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (34,-33).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218737-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{34, -33}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218736(n):=(33^n-1)/32$
makelist(A218736(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218736(n)=33^n>>5
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
E.g.f.: exp(x)*(exp(32*x) - 1)/32. - Stefano Spezia, Mar 24 2023

A066443 Number of distinct paths of length 2n+1 along edges of a unit cube between two fixed adjacent vertices.

Original entry on oeis.org

1, 7, 61, 547, 4921, 44287, 398581, 3587227, 32285041, 290565367, 2615088301, 23535794707, 211822152361, 1906399371247, 17157594341221, 154418349070987, 1389765141638881, 12507886274749927, 112570976472749341

Offset: 0

John W. Layman, Aug 12 2002

All members of sequence are also hex, or central hexagonal, numbers (A003215). (If n is a hex number, 9n - 2 is always a hex number; see recurrence.) - Matthew Vandermast, Mar 30 2003
The sequence 1,1,7,61,547,... with g.f. (1-9x+6x^2)/((1-x)(1-9x)) and a(n) = A054879(n)/3 + 2*0^n/3 gives the denominators in the probability that a random walk on the cube returns to its starting corner on the 2n-th step. - Paul Barry, Mar 11 2004
Equals row sums of even row terms of triangle A158303. - Gary W. Adamson, Mar 15 2009
It appears that a(n) is the n-th record value in A120437, which gives the  differences of A037314 (positive integers n such that the sum of the base 3 digits of n equals the sum of the base 9 digits of n). - John W. Layman, Dec 14 2010
Numbers in base 9 are 1, 6+1, 66+1, 666+1, 6666+1, 66666+1, etc.; that is, n 6's + 1. - Yuchun Ji, Aug 15 2019
All prime factors of a(n) are 1 mod 6. In addition, if n is not 1 mod 3 (first index being n=0), then 3 is a cubic residue modulo all prime factors of a(n). This provides a simple proof that there are infinitely many primes 1 mod 6 that have 3 as a cubic residue. - William Hu, Jul 26 2024

			From _Michael B. Porter_, Aug 22 2016: (Start)
Give coordinates (a,b,c) to the vertices of the cube, where a, b, and c are either 0 or 1. For n = 1, the a(1) = 7 paths of length 2n + 1 = 3 from (0,0,0) to (0,0,1) are:
(0,0,0) -> (0,0,1) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (0,0,1) -> (1,0,1) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (0,1,0) -> (0,1,1) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (0,0,0) -> (0,0,1)
(0,0,0) -> (1,0,0) -> (1,0,1) -> (0,0,1) (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417.
E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - _N. J. A. Sloane_, Feb 28 2013
E. Estrada and J. A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54:369-391, 1947.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A158303, A037314, A120437, A083234 (binomial transform), A083233 (inverse binomial transform), A054879 (recurrent walks), A125857 (walks ending on face diagonal), A054880 (walks ending on space diagonal).

[(3^(2*n+1)+1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

seq((3^(2*n+1) + 1)/4, n=0..18); # Zerinvary Lajos, Jun 16 2007

NestList[9 # - 2 &, 1, 18] (* or *)
Table[(3^(2 n + 1) + 1)/4, {n, 0, 18}] (* or *)
CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 9 x)), {x, 0, 18}], x] (* Michael De Vlieger, Aug 22 2016 *)

a(n)=3^(2*n+1)\/4 \\ Charles R Greathouse IV, Jul 02 2013

Vec((1-3*x)/((1-x)*(1-9*x)) + O(x^50)) \\ Altug Alkan, Nov 13 2015

a(n) = (3^(2*n+1)+1)/4. - Vladeta Jovovic, Dec 22 2002
a(n) = 9*a(n-1) - 2. - Matthew Vandermast, Mar 30 2003
From Paul Barry, Apr 21 2003: (Start)
G.f.: (1-3*x)/((1-x)*(1-9*x)).
E.g.f.: (3*exp(9*x) + exp(x))/4. (End)
a(n) = (-1)^n times the (i, i)-th element of M^n (for any i), where M = ((1, 1, 1, -2), (1, 1, -2, 1), (1, -2, 1, 1), (-2, 1, 1, 1)). - Simone Severini, Nov 25 2004
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*4^(n-k). - Paul Barry, Jan 22 2005
a(n) = A054880(n) + 1.
a(n) = A057660(3^n). - Henry Bottomley, Nov 08 2015
a(n) = Sum_{k=0..2n} (-3)^k == 1 + Sum_{k=1..n} 2*3^(2k-1). - Bob Selcoe, Aug 21 2016
a(n) = 3^(2*n+1) * a(-1-n) for all n in Z. - Michael Somos, Jul 02 2017
a(n) = 6*A002452(n) + 1. - Yuchun Ji, Aug 15 2019

Corrected by Vladeta Jovovic, Dec 22 2002

A125118 Triangle read by rows: T(n,k) = value of the n-th repunit in base (k+1) representation, 1<=k<=n.

Original entry on oeis.org

1, 3, 4, 7, 13, 21, 15, 40, 85, 156, 31, 121, 341, 781, 1555, 63, 364, 1365, 3906, 9331, 19608, 127, 1093, 5461, 19531, 55987, 137257, 299593, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 511, 9841, 87381, 488281, 2015539, 6725601, 19173961, 48427561, 111111111

Offset: 1

Reinhard Zumkeller, Nov 21 2006

			First 4 rows:
1: [1]_2
2: [11]_2 ........ [11]_3
3: [111]_2 ....... [111]_3 ....... [111]_4
4: [1111]_2 ...... [1111]_3 ...... [1111]_4 ...... [1111]_5
_
1: 1
2: 2+1 ........... 3+1
3: (2+1)*2+1 ..... (3+1)*3+1 ..... (4+1)*4+1
4: ((2+1)*2+1)*2+1 ((3+1)*3+1)*3+1 ((4+1)*4+1)*4+1 ((5+1)*5+1)*5+1.

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Repunit

This triangle shares some features with triangle A104878.
This triangle is a portion of rectangle A055129.
Each term of A110737 comes from the corresponding row of this triangle.
Diagonals (adjusting offset as necessary): A060072, A023037, A031973, A173468.
Columns (adjusting offset as necessary): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
Cf. A023037, A031973, A125119, A125120 (row sums).

[((k+1)^n -1)/k : k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 15 2022

Table[((k+1)^n -1)/k, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Aug 15 2022 *)

def A125118(n,k): return ((k+1)^n -1)/k
flatten([[A125118(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Aug 15 2022

T(n, k) = Sum_{i=0..n-1} (k+1)^i.
T(n+1, k) = (k+1)*T(n, k) + 1.
Sum_{k=1..n} T(n, k) = A125120(n).
T(2*n-1, n) = A125119(n).
T(n, 1) = A000225(n).
T(n, 2) = A003462(n) for n>1.
T(n, 3) = A002450(n) for n>2.
T(n, 4) = A003463(n) for n>3.
T(n, 5) = A003464(n) for n>4.
T(n, 9) = A002275(n) for n>8.
T(n, n) = A060072(n+1).
T(n, n-1) = A023037(n) for n>1.
T(n, n-2) = A031973(n) for n>2.
T(n, k) = A055129(n, k+1) = A104878(n+k, k+1), 1<=k<=n. - Mathew Englander, Dec 19 2020

A338090 Numbers having at least one 8 in their representation in base 9.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 89, 98, 107, 116, 125, 134, 143, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 170, 179, 188, 197, 206, 215, 224, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 251, 260, 269, 278, 287, 296, 305, 314, 315

Offset: 1

François Marques, Oct 09 2020

Blocks of consecutive terms have lengths in A002452. - Devansh Singh, Oct 21 2020

			70 is not in the sequence since it is 77_9 in base 9, but 76 is in the sequence since it is 84_9 in base 9.

François Marques, Table of n, a(n) for n = 1..10000

Cf. A007095 (base 9).
Complement of A037477.
Cf. A043485 (numbers with exactly one 8 in base 9).
Cf. Numbers with at least one digit b-1 in base b: A074940 (b=3), A337250 (b=4), A337572 (b=5), A333656 (b=6), A337141 (b=7), A337239 (b=8), this sequence (b=9), A011539 (b=10), A095778 (b=11).
Cf. Numbers with no digit b-1 in base b: A005836 (b=3), A023717 (b=4), A020654 (b=5), A037465 (b=6), A020657 (b=7), A037474 (b=8), A037477 (b=9), A007095 (b=10), A171397 (b=11).

seq(`if`(numboccur(8, convert(n, base, 9))>0, n, NULL), n=0..100);

Select[ Range[ 0, 100 ], (Count[ IntegerDigits[ #, 9 ], 8 ]>0)& ]

isok(m) = #select(x->(x==8), digits(m, 9)) >= 1;

from gmpy2 import digits
def A338090(n):
    def f(x):
        l = (s:=digits(x,9)).find('8')
        if l >= 0: s = s[:l]+'7'*(len(s)-l)
        return n+int(s,8)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Dec 04 2024

A014827 a(1)=1, a(n) = 5*a(n-1) + n.

Original entry on oeis.org

1, 7, 38, 194, 975, 4881, 24412, 122068, 610349, 3051755, 15258786, 76293942, 381469723, 1907348629, 9536743160, 47683715816, 238418579097, 1192092895503, 5960464477534, 29802322387690, 149011611938471, 745058059692377, 3725290298461908, 18626451492309564, 93132257461547845

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), pp. 461-469.
Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016218, A016256, A023000, A023001.

[(5^(n+1)-4*n-5)/16: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

a:=n->sum((5^(n-j)-1^(n-j))/4,j=0..n): seq(a(n), n=1..21); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=7},Table[c=6*b-5*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

[(gaussian_binomial(n,1,5)-n)/4 for n in range(2,23)] # Zerinvary Lajos, May 29 2009

a(n) = (5^(n+1) - 4*n - 5)/16.
G.f.: x/((1-5*x)*(1-x)^2).
From Paul Barry, Jul 30 2004: (Start)
a(n) = Sum_{k=0..n} (n-k)*5^k = Sum_{k=0..n} k*5^(n-k).
a(n) = Sum_{k=0..n} binomial(n+2,k+2)*4^k [Offset 0]. (End)
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: exp(x)*(5*exp(4*x) - 4*x - 5)/16.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 3. (End)

A104878 A sum-of-powers number triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 15, 13, 5, 1, 1, 6, 31, 40, 21, 6, 1, 1, 7, 63, 121, 85, 31, 7, 1, 1, 8, 127, 364, 341, 156, 43, 8, 1, 1, 9, 255, 1093, 1365, 781, 259, 57, 9, 1, 1, 10, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 1, 11, 1023, 9841, 21845

Offset: 0

Paul Barry, Mar 28 2005

Columns are partial sums of the columns of A004248. Row sums are A104879. Diagonal sums are A104880.

The rows of this triangle (apart from the initial "1" in each row) are the antidiagonals of rectangle A055129. The diagonals of this triangle (apart from the initial "1") are the rows of rectangle A055129. The columns of this triangle (apart from the leftmost column) are the same as the columns of rectangle A055129 but shifted downward. - Mathew Englander, Dec 21 2020

			Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 15, 13,  5,  1;
  1,  6, 31, 40, 21,  6,  1;
  ...

Cf. A004248 (first differences by column), A104879 (row sums), A104880 (antidiagonal sums), A125118 (version of this triangle with fewer terms).
This triangle (ignoring the leftmost column) is a rotation of rectangle A055129.
Columns (adjusting offset as necessary): A000012, A000027, A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724, A218725, A218726, A218727, A218728, A218729, A218730, A218731, A218732, A218733, A218734, A132469, A218736, A218737, A218738, A218739, A218740, A218741, A218742, A218743, A218744, A218745, A218746, A218747, A218748, A218749, A218750, A218751, A218753, A218752.
T(2n,n) gives A031973.

A104878 :=proc(n,k): if k = 0 then 1 elif k=1 then n elif k>=2 then (k^(n-k+1)-1)/(k-1) fi: end: for n from 0 to 7 do seq(A104878(n,k), k=0..n) od; seq(seq(A104878(n,k), k=0..n), n=0..10); # Johannes W. Meijer, Aug 21 2011

T(n, k) = if(k=1, n, if(k<=n, (k^(n-k+1)-1)/(k-1), 0));
G.f. of column k: x^k/((1-x)(1-k*x)). [corrected by Werner Schulte, Jun 05 2019]
T(n, k) = A069777(n+1,k)/A069777(n,k). [Johannes W. Meijer, Aug 21 2011]
T(n, k) = A055129(n+1-k, k) for n >= k > 0. - Mathew Englander, Dec 19 2020

A015008 q-factorial numbers for q=9.

Original entry on oeis.org

1, 1, 10, 910, 746200, 5507702200, 365876657146000, 218747042884536166000, 1177042838234827583459440000, 57001313848230245122464621625840000, 24843911488189148287648216529610193612000000, 97453533413342456299179976631323547842824103012000000

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015009, A015011.
Column q=9 of A069777.
Cf. A002452, A132037.

[n le 1 select 1 else (9^n - 1)*Self(n-1)/8: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((9^n - 1) * a[n-1])/8}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 9], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (9^k - 1) / (9 - 1).
a(0) = 1, a(n) = (9^n - 1)*a(n-1)/8. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A002452(k).
a(n) ~ c * 3^(n*(n+1))/8^n, where c = A132037. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A052386 Number of integers from 1 to 10^n-1 that lack 0 as a digit.

Original entry on oeis.org

0, 9, 90, 819, 7380, 66429, 597870, 5380839, 48427560, 435848049, 3922632450, 35303692059, 317733228540, 2859599056869, 25736391511830, 231627523606479, 2084647712458320, 18761829412124889, 168856464709124010, 1519708182382116099, 13677373641439044900

Offset: 0

Odimar Fabeny, Mar 10 2000

			For n=2, the numbers from 1 to 99 which *have* 0 as a digit are the 9 numbers 10, 20, 30, ..., 90. So a(1) = 99 - 9 = 90.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A024101, A052379.

Row n=9 of A228275.

[9*(9^n-1)/8: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

Table[9(9^n - 1)/8, {n, 0, 20}]
LinearRecurrence[{10,-9},{0,9},30] (* Harvey P. Dale, Mar 22 2019 *)

a(n)=9^(n+1)\8 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = 9*a(n-1) + 9.
a(n) = 9*(9^n-1)/8 = sum_{j=1..n} 9^j = a(n-1)+9^n = 9*A002452(n) = A002452(n+1)-1; write A000918(n+1) in base 2 and read as if written in base 9. - Henry Bottomley, Aug 30 2001
a(n) = 10*a(n-1)-9*a(n-2). G.f.: 9*x / ((x-1)*(9*x-1)). - Colin Barker, Sep 26 2013

More terms and revised description from James Sellers, Mar 13 2000
More terms and revised description from Robert G. Wilson v, Apr 14 2003
More terms from Colin Barker, Sep 26 2013

A138894 Expansion of (1+x)/(1-10x+9x^2).

Original entry on oeis.org

1, 11, 101, 911, 8201, 73811, 664301, 5978711, 53808401, 484275611, 4358480501, 39226324511, 353036920601, 3177332285411, 28595990568701, 257363915118311, 2316275236064801, 20846477124583211, 187618294121248901

Offset: 0

Paul Barry, Apr 02 2008

Orbit starting at 1 of A138893: a(n)=A138893^(n)(1). Partial sums of A003952.

Sum of n-th row of triangle of powers of 9: 1; 1 9 1; 1 9 81 9 1; 1 9 81 729 81 9 1; ... - Philippe Deléham, Feb 22 2014

			a(0) = 1;
a(1) = 1 + 9 + 1 = 11;
a(2) = 1 + 9 + 81 + 9 + 1 = 101;
a(3) = 1 + 9 + 81 + 729 + 81 + 9 + 1 = 911; etc. - _Philippe Deléham_, Feb 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A096053 ((3*9^n-1)/2), a(n+1)=9a(n)-4 in A135423.

Cf. A002452, A003952, A135522, A138893.

[(5/4)*9^n-1/4: n in [0..20]]; // Vincenzo Librandi, Aug 09 2011

Table[(5*9^n - 1)/4, {n, 0, 18}] (* L. Edson Jeffery, Feb 13 2015 *)

Vec((1+x)/(1-10*x+9*x^2) + O(x^30)) \\ Michel Marcus, Feb 13 2015

G.f.: (1+x)/((1-x)*(1-9x)).
a(n) = (5/4)*9^n - 1/4.
a(n) = A002452(n) + A002452(n+1).
Bisection of A135522/3. a(n+1)=9*a(n)+2. - Paul Curtz, Apr 22 2008
a(n) = Sum_{k=0..n} A112468(n,k)*10^k. - Philippe Deléham, Feb 22 2014

A008958 Triangle of central factorial numbers 4^k T(2n+1, 2n+1-2k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 35, 91, 1, 1, 84, 966, 820, 1, 1, 165, 5082, 24970, 7381, 1, 1, 286, 18447, 273988, 631631, 66430, 1, 1, 455, 53053, 1768195, 14057043, 15857205, 597871, 1, 1, 680, 129948, 8187608, 157280838, 704652312, 397027996, 5380840, 1

Offset: 0

N. J. A. Sloane

			From _Wesley Transue_, Jan 21 2012: (Start)
Triangle begins:
  1;
  1,   1;
  1,  10,      1;
  1,  35,     91,       1;
  1,  84,    966,     820,         1;
  1, 165,   5082,   24970,      7381,         1;
  1, 286,  18447,  273988,    631631,     66430,         1;
  1, 455,  53053, 1768195,  14057043,  15857205,    597871,       1;
  1, 680, 129948, 8187608, 157280838, 704652312, 397027996, 5380840, 1;
(End)

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Robert James Purser, Mobius Net Cubed-Sphere Gnomonic Grids, U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Weather Service, National Centers for Environmental Protection, 2018.

Cf. A008955-A008957, A036969.

Columns include A000447. Right-hand columns include A002452, A002453.

Flatten[Table[Sum[(-1)^(q+1) 4^(p-n) (2p+2q-2n-1)^(2n+1)/((2n+1-2p-q)! q!), {q, 0, n-p}], {n, 0, 8}, {p, 0, n}]] (* Wesley Transue, Jan 21 2012 *)

G.f. of i-th right-hand column is x/Product_{j=1..i+1} (1 - (2j-1)^2*x).

More terms from Vladeta Jovovic, Apr 16 2000

cp's OEIS Frontend

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