A018387 -id:A018387 - cp's OEIS frontend

A125750 A Moessner triangle using (1, 3, 5, ...).

1, 3, 5, 10, 19, 11, 42, 89, 64, 19, 216, 498, 415, 160, 29, 1320, 3254, 3023, 1385, 335, 41, 9360, 24372, 24640, 12803, 3745, 623, 55, 75600, 206100, 223116, 127799, 42938, 8750, 1064, 71, 685440, 1943568, 2227276, 1380076, 516201, 122010, 18354, 1704

Offset: 1

Gary W. Adamson, Dec 06 2006

Right border of the triangle = A028387, left border = A007680.

			Circling the 1, 3, 6, ...(-th) terms in the sequence (1, 3, 5, 7, ...), we get A018387: (1, 5, 11, 19, 29, ...). Taking partial sums of the remaining terms, we get (3, 10, 19, 32, ...) in row 2 and we circle 3 and 19. In row 3 we circle the 10.
First few rows of the triangle are:
    1;
    3,   5;
   10,  19,  11;
   42,  89,  64,  19;
  216, 498, 415, 160,  29;
  ...

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, pp. 63-64.

Joshua Zucker, Table of n, a(n) for n = 1..78
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A125714, A125751, A125752.

Using "Moessner's Magic" (Conway and Guy, pp. 63-64; cf. A125714), we circle the 1, 3, 6, 10, ...(-th) terms in the sequence (1, 3, 5, 7, ...) and take partial sums of the remaining terms, making row 2. Circle the terms in row 2 one place offset to the left of row 1 terms, then take partial sums. Continue with analogous operations for succeeding rows. The triangle = leftmost circled terms in each row.

More terms from Joshua Zucker, Jun 17 2007

A132787 Triangle read by rows: T(n,k) = 2*A001263(n,k) - 1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 39, 19, 1, 1, 29, 99, 99, 29, 1, 1, 41, 209, 349, 209, 41, 1, 1, 55, 391, 979, 979, 391, 55, 1, 1, 71, 671, 2351, 3527, 2351, 671, 71, 1, 1, 89, 1079, 5039, 10583, 10583, 5039, 1079, 89, 1, 1, 109, 1649, 9899, 27719, 38807, 27719, 9899, 1649, 109, 1

Offset: 1

Gary W. Adamson, Aug 30 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 5, 1;
  1, 11, 11, 1;
  1, 19, 39, 19, 1;
  1, 29, 99, 99, 29, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Column 2 is A018387.
Row sums are A132788.
Cf. A001263.

T(n,k) = if(k<=n, 2*binomial(n-1, k-1) * binomial(n, k-1) / k - 1, 0); \\ Andrew Howroyd, Aug 10 2018

Equals 2*A001263 - A000012 as infinite lower triangular matrices; where A001263 = the Narayana triangle.

T(n,k) = 2*binomial(n-1, k-1) * binomial(n, k-1) / k - 1. - Andrew Howroyd, Aug 10 2018

a(20) corrected and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A130299 A130296 * A051340.

Original entry on oeis.org

1, 3, 2, 5, 3, 3, 7, 4, 4, 4, 9, 5, 5, 5, 5, 11, 6, 6, 6, 6, 6, 13, 7, 7, 7, 7, 7, 7, 15, 8, 8, 8, 8, 8, 8, 8, 17, 9, 9, 9, 9, 9, 9, 9, 9, 19, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A018387: (1, 5, 11, 19, 29, 41, 55, ...).

A130298 = A051340 * A130296.

			First few rows of the triangle:
   1;
   3, 2;
   5, 3, 3;
   7, 4, 4, 4;
   9, 5, 5, 5, 5;
  11, 6, 6, 6, 6, 6;
  13, 7, 7, 7, 7, 7, 7;
  ...

Cf. A051340, A130296, A130298, A028387.

A130296 * A051340 as infinite lower triangular matrices.

cp's OEIS Frontend

A125750 A Moessner triangle using (1, 3, 5, ...).

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A132787 Triangle read by rows: T(n,k) = 2*A001263(n,k) - 1.

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A130299 A130296 * A051340.

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