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A243821 Three-column table read by rows: coefficient table for generalized m-nomial transforms of Fibonacci numbers (A000045). 0
2, 1, 0, 5, 1, 1, 9, 4, 2, 16, 5, 4, 27, 11, 7, 45, 16, 12, 74, 29, 20, 121, 45, 33, 197, 76, 54, 320, 121, 88, 519, 199, 143, 841, 320, 232, 1362, 521, 376, 2205, 841, 609, 3569, 1364, 986 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

GENERAL EXPLANATION OF TERMS AND CONCEPTS: (Start)

I.  Let the term "transform" generally mean the operation of summing the products of the ascending numbers of a sequence and numbers in the j-th row of an m-nomial triangle (i.e., triangle of m-nomial coefficients, T(j,k)), thus generating a new sequence for each m as j increases {j=0..inf.}. Further, let T(0,0) be the top entry (0th row, 0th column) in an m-nomial triangle, where m >= 2 (m=2 being Pascal's triangle, m=3 the trinomial triangle, etc.) and {k=0..(m-1)*j} for row j.

II.  In this entry, the sequence is the Fibonacci numbers (A000045), starting with F(1)=1. So for example, the pentanomial (m=5) transform of the Fibonacci numbers is {1, 12, 193, 3149, 51410, ...} because {1*1} = 1 when j=0 (transform of row 0); {1*1 + 1*1 + 1*2 + 1*3 + 1*5} = 12 when j=1 (transform of row 1); {1*1 + 2*1 + 3*2 + 4*3 + 5*5 + 4*8 + 3*13 + 2*21 + 1*34} = 193 when j=2 (transform of row 2), ad infinitum.

III.  Let T'<m*> denote the entire sequence for the m-nomial transform of the Fibonacci numbers.  So for example, T'<4*> is {1, 7, 65, 615, 5825, ...). Further, let T'<m>(n'') denote the transform at n''. So for example, when n''=3: T'<4>(3)=615.

IV.  Every m-nomial transform of Fibonacci numbers can be generated using the table below (presented in three columns: a'', b'', and c''; see "Example" section) and its expansion, both with and without internal recurrence (see "Formula" section for expanded form).

V.  Coefficients a'', b'' and c'' for row m (corresponding to the m-nomial triangle, the initial row being m=2) are used to find T'<m*> and T'<m>(n''), applying the following equation:

VI.  GENERAL M-NOMIAL TRANSFORM EQUATION: T'<m>(n'') = a''*T'<m>(n''-1) + b''*(sum_T'<m>(i)) {i=0..(n''-2); n''>=2} - c''.

(Given: T'<m>(0)=1)

See "Example" section for T'<5*> (pentanomial transform: using row m=5 in table for a'', b'' and c''). (End)

LINKS

Table of n, a(n) for n=0..44.

FORMULA

I. EXPANSION FORMULAS without internal recurrence [corresponding table columns noted]:

a(n) = A000032(m+1)-2 when n == 0 (mod 3) [corresponds to column a'' in table].

a(n) = A000032(m-1) when n == 1 (mod 6) [column b'', even m].

a(n) = A000032(m-1)-2 when n == 4 (mod 6) [column b'', odd m].

a(n) = A000045(m)-1 when n == 2 (mod 3) [column c''].

II. EXPANSION FORMULAS with internal recurrence:

a(n) = 2*a(n-3) + a(n-9), n >= 9, when n == 0 (mod 3) [corresponds to column a'' in table].

a(n) = a(n-4) - a(n-3), n >= 4, when n == 1 (mod 3) [column b''].

a(n) = A000045(m)-1 when n == 2 (mod 3) [column c''].

EXAMPLE

I.  COEFFICIENT TABLE (initial row: m=2)

   m       a"    b"    c"

   2       2     1     0

   3       5     1     1

   4       9     4     2

   5      16     5     4

   6      27    11     7

   7      45    16    12

   8      74    29    20

   9     121    45    33

  10     197    76    54

  11     320   121    88

  12     519   199   143

  13     841   320   232

  14    1362   521   376

II.  EXAMPLES OF EXPANSION FORMULAS for a(n):

a(12) = 27: (m=6; 12 == 0 (mod 3). A000032(7)-2 = 27, 2*16-5 = 27).

a(19) = 29: (m=8; 19 == 1 (mod 6). A000032(7) = 29, 45-16 = 29).

a(22) = 45: (m=9; 22 == 4 (mod 6). A000032(8)-2 = 45, 74-29 = 45).

a(23) = 33: (m=9; 23 == 2 (mod 3).  A000045(9)-1 = 33).

III.   EXAMPLE OF M-NOMIAL TRANSFORM USING GENERAL EQUATION, FOR T'<5*> (pentanomial transform -- m=5; a''=16, b''=5, c''=4. See "Comments" for explanation):

T'<5>(0) = 1 (given);

T'<5>(1) = 12:  [16*T'<5>(0) - 4] = [16*1 - 4] = 12.

T'<5>(2) = 193:  [16*T'<5>(1) + 5*T'<5>(0) - 4] = [16*12 + 5*1 - 4] = 193.

T'<5>(3) = 3149:  [16*T'<5>(2) + 5*(T'<5>(1) + T'<5>(0)) - 4] = [16*193 + 5*(12+1) - 4] = 3149.

T'<5>(4) = 51410:  [16*T'<5>(3) + 5*(T'<5>(2) + T'<5>(1) + T'<5>(0)) - 4] = [16*3149 + 5*(193+12+1) - 4] = 51410.

Therefore T'<5*> = {1, 12, 193, 3149, 51410, ...}.

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Cf. A007318 (Pascal's triangle), A027907 (trinomial triangle).

Sequence in context: A112334 A113469 A060137 * A143445 A133727 A103185

Adjacent sequences:  A243818 A243819 A243820 * A243822 A243823 A243824

KEYWORD

nonn

AUTHOR

Bob Selcoe, Jun 11 2014

STATUS

approved

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Last modified September 19 11:00 EDT 2019. Contains 327192 sequences. (Running on oeis4.)