A010048 -id:A010048

A001657

Fibonomial coefficients: column 5 of A010048.
(Formerly M4568 N1945)

+20
8

1, 8, 104, 1092, 12376, 136136, 1514513, 16776144, 186135312, 2063912136, 22890661872, 253854868176, 2815321003313, 31222272414424, 346260798314872, 3840089017377228, 42587248616222024, 472299787252290712, 5237885063192296801, 58089034826620525728

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FORMULA

a(n) = A010048(5+n, 5) (or fibonomial(5+n, 5)).

CROSSREFS

Cf. A010048, A001654-A001658, A065563.

A056569

Row sums of Fibonomial triangle A010048.

+20
7

1, 2, 3, 6, 14, 42, 158, 756, 4594, 35532, 349428, 4370436, 69532964, 1407280392, 36228710348, 1186337370456, 49415178236344, 2618246576596392, 176462813970065208, 15128228719573952976, 1649746715671916095304

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = Sum_{m=0..n} A010048(n, m), where A010048(n, m) = fibonomial(n, m).

CROSSREFS

Cf. A010048, A062073, A181926.

A181926

Diagonal sums of Fibonomial triangle A010048.

+20
4

1, 1, 2, 2, 4, 6, 13, 27, 70, 191, 609, 2130, 8526, 38156, 194000, 1109673, 7176149, 52238676, 429004471, 3970438003, 41454181730, 488046132076, 6482590679282, 97134793638750, 1641654359781521, 31285014253070731, 672372121341768918, 16299021330860540657

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A003266, A010048, A056569, A062073.

A103910

Triangle T read by rows: inverse of fibonomial triangle (A010048).

+20
1

1, -1, 1, 0, -1, 1, 1, 0, -2, 1, -1, 3, 0, -3, 1, -6, -5, 15, 0, -5, 1, 35, -48, -40, 60, 0, -8, 1, 181, 455, -624, -260, 260, 0, -13, 1, -6056, 3801, 9555, -6552, -1820, 1092, 0, -21, 1, -3741, -205904, 129234, 162435, -74256, -12376, 4641, 0, -34, 1

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FORMULA

Conjecture: T(n+k, n) = A010048(n+k-1, k)*T(k, 1), n>1.

a(n,k) = A010048(n,k) * (Sum[s=1..n-k;(-1)^s * Sum[k1+k2+..+ks=n-k,ki>=1; C(n-k; k1,k2,...,ks)] ]) where C(n; k1,k2,...,ks) is a multi-F-nomial coefficient. - Maciej Dziemianczuk, Dec 21 2008

A172373

Triangle T(n, k) = A010048(n, k)*A010048(n, k-1)/Fibonacci(n), read by rows.

+20
1

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 15, 45, 15, 1, 1, 40, 300, 300, 40, 1, 1, 104, 2080, 5200, 2080, 104, 1, 1, 273, 14196, 94640, 94640, 14196, 273, 1, 1, 714, 97461, 1689324, 4504864, 1689324, 97461, 714, 1, 1, 1870, 667590, 30375345, 210602392, 210602392, 30375345, 667590, 1870, 1

(list; graph; refs; listen; history; text; internal format)

FORMULA

T(n, k) = A010048(n, k) * A010048(n, k-1) / Fibonacci(n) where A010048 are the Fibonomial coefficients. - Michel Marcus, Oct 23 2019

MATHEMATICA

A010048[n_, k_]:= Product[Fibonacci[n-j+1]/Fibonacci[j], {j, k}];

T[n_, k_]:= A010048[n, k]*A010048[n, k-1]/Fibonacci[n];

PROG

(PARI) fibonomial(n, k) = prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)); \\ A010048

def A010048(n, q): return product( fibonacci(n-j+1)/fibonacci(j) for j in (1..k) )

def T(n, k, q): return A010048(n, k)*A010048(n, k-1)/fibonacci(n)

CROSSREFS

Cf. A010048 (Fibonomial coefficients), A001263 (Narayana numbers).

A181927

Row square-sums of Fibonomial triangle A010048.

+20
1

1, 2, 3, 10, 56, 502, 6930, 157172, 5847270, 350430420, 33789991248, 5280020814732, 1338210835193414, 548265785425359340, 363248986031094300018, 389399454403643525265020, 675824289510077938157099920

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A010048, A056569.

A001654

Golden rectangle numbers: F(n)*F(n+1), where F(n) = A000045(n) (Fibonacci numbers).
(Formerly M1606 N0628)

+10
122

0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895, 12816, 33552, 87841, 229970, 602070, 1576239, 4126648, 10803704, 28284465, 74049690, 193864606, 507544127, 1328767776, 3478759200, 9107509825, 23843770274, 62423800998, 163427632719

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LINKS

Dale Gerdemann, Golden Ratio Base Digit Patterns for Columns of the Fibonomial Triangle, "Another interesting pattern is for Golden Rectangle Numbers A001654. I made a short video illustrating this pattern, along with other columns of the Fibonomial Triangle A010048".

FORMULA

a(n) = A010048(n+1, 2) = Fibonomial(n+1, 2).

CROSSREFS

Cf. A001655, A001656, A001657, A001658, A010048, A067962.

A055870

Signed Fibonomial triangle.

+10
27

1, 1, -1, 1, -1, -1, 1, -2, -2, 1, 1, -3, -6, 3, 1, 1, -5, -15, 15, 5, -1, 1, -8, -40, 60, 40, -8, -1, 1, -13, -104, 260, 260, -104, -13, 1, 1, -21, -273, 1092, 1820, -1092, -273, 21, 1, 1, -34, -714, 4641, 12376, -12376, -4641, 714, 34, -1, 1, -55, -1870, 19635, 85085, -136136, -85085, 19635, 1870, -55, -1

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COMMENTS

The inverse of the row polynomial p(n,x) := Sum_{m=0..n} T(n,m)*x^m is the g.f. for the column m=n-1 of the Fibonomial triangle A010048.

FORMULA

T(n, m) = (-1)^floor((m+1)/2)*A010048(n, m), where A010048(n, m) := fibonomial(n, m).

EXAMPLE

Row polynomial for n=4: p(4,x) = 1-3*x-6*x^2+3*x^3+x^4 = (1+x-x^2)*(1-4*x-x^2). 1/p(4,x) is G.f. for A010048(n+3,3), n >= 0: {1,3,15,60,...} = A001655(n).

MAPLE

(-1)^floor((k+1)/2)*A010048(n, k) ;

CROSSREFS

Cf. A000032, A000045, A001654, A001655, A001656, A001657, A001658, A010048, A051159, A056565, A056566, A056567.

A256890

Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

+10
24

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

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COMMENTS

If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).

A056588

Coefficient triangle of certain polynomials.

+10
18

1, 1, -1, 1, -2, -1, 1, -4, -4, 1, 1, -7, -16, 7, 1, 1, -12, -53, 53, 12, -1, 1, -20, -166, 318, 166, -20, -1, 1, -33, -492, 1784, 1784, -492, -33, 1, 1, -54, -1413, 9288, 17840, -9288, -1413, 54, 1, 1, -88, -3960, 46233, 163504, -163504, -46233, 3960, 88, -1

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FORMULA

Sum_{j=0..n-1} a(n-1, n-1-j)*A010048(k+j, n) = Fibonacci(k)^n. - Tony Foster III, Jul 24 2018