Search: a060690 -id:a060690
|
| |
|
|
A140051
|
|
L.g.f.: A(x) = log(G(x)) where G(x) = g.f. of A060690(n) = C(2^n+n-1,n).
|
|
+20
1
|
|
|
|
2, 16, 308, 14488, 1843232, 714580528, 917085102992, 4076698622618144, 64300718807613519968, 3649606003781552269341376, 752497581806524062754828125952, 567745591696108934746387351412913664
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
LINKS
|
|
|
|
FORMULA
|
L.g.f.: A(x) = log[ Sum_{n>=0} (-log(1 - 2^n*x))^n/n! ].
|
|
|
EXAMPLE
|
A(x) = 2*x + 16*x^2/2 + 308*x^3/3 + 14488*x^4/4 + 1843232*x^5/5 +...
A(x) = log(G(x)) where G(x) = g.f. of A060690:
G(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 +... + C(2^n+n-1,n)*x^n +...
|
|
|
PROG
|
(PARI) {a(n)=n*polcoeff(log(sum(k=0, n, binomial(2^k+k-1, k)*x^k)+x*O(x^n)), n)}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A014070
|
|
a(n) = binomial(2^n, n).
|
|
+10
47
|
|
|
|
1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, 409663695276000, 6208116950265950720, 334265867498622145619456, 64832175068736596027448301568, 45811862025512780638750907861652480, 119028707533461499951701664512286557511680
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
a(n) is the number of n X n (0,1) matrices with distinct rows modulo rows permutations. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!. - Paul D. Hanna, Dec 28 2007
a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n, k) * 2^(n*k). - Paul D. Hanna, Feb 05 2023
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
(End)
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) a(n)=binomial(2^n, n)
(PARI) /* G.f. A(x) as Sum of Series: */
a(n)=polcoeff(sum(k=0, n, log(1+2^k*x +x*O(x^n))^k/k!), n) \\ Paul D. Hanna, Dec 28 2007
(PARI) {a(n) = (1/n!) * sum(k=0, n, stirling(n, k, 1) * 2^(n*k) )}
(Sage) [binomial(2^n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
|
|
|
CROSSREFS
|
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), this sequence (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A136556
|
|
a(n) = binomial(2^n - 1, n).
|
|
+10
27
|
|
|
|
1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, 396861704798625, 6098989894499557055, 331001552386330913728641, 64483955378425999076128999167, 45677647585984911164223317311276545, 118839819203635450208125966070067352769535, 1144686912178270649701033287538093722740144666625
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.
Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:
{1}{2}{3} {1}{2}{12} {1}{2}{123} {1}{12}{123} {12}{13}{123}
{1}{2}{13} {1}{3}{123} {1}{13}{123} {12}{23}{123}
{1}{2}{23} {1}{12}{13} {1}{23}{123} {13}{23}{123}
{1}{3}{12} {1}{12}{23} {2}{12}{123}
{1}{3}{13} {1}{13}{23} {2}{13}{123}
{1}{3}{23} {2}{3}{123} {2}{23}{123}
{2}{3}{12} {2}{12}{13} {3}{12}{123}
{2}{3}{13} {2}{12}{23} {3}{13}{123}
{2}{3}{23} {2}{13}{23} {3}{23}{123}
{3}{12}{13} {12}{13}{23}
{3}{12}{23}
{3}{13}{23}
Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.
(End)
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2^n,k).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.
G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Length[Subsets[Rest[Subsets[Range[n]]], {n}]], {n, 0, 4}] (* Gus Wiseman, Dec 19 2023 *)
|
|
|
PROG
|
(PARI) {a(n) = binomial(2^n-1, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As coefficient of x^n in the g.f.: */
{a(n) = polcoeff( sum(i=0, n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))
(Sage) [binomial(2^n -1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n -1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
(Python)
from math import comb
|
|
|
CROSSREFS
|
Sequences of the form binomial(2^n +p*n +q, n): this sequence (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A016121
|
|
Number of sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.
|
|
+10
17
|
|
|
|
1, 2, 5, 17, 86, 698, 9551, 226592, 9471845, 705154187, 94285792211, 22807963405043, 10047909839840456, 8110620438438750647, 12062839548612627177590, 33226539134943667506533207, 170288915434579567358828997806, 1630770670148598007261992936663653
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Number of n X n binary symmetric matrices with rows, considered as binary numbers, in nondecreasing order. - R. H. Hardin, May 30 2008
Also, number of (n+1) X (n+1) binary symmetric matrices with zero main diagonal and rows, considered as binary numbers, in nondecreasing order. - Max Alekseyev, Feb 06 2022
|
|
|
LINKS
|
|
|
|
FORMULA
|
Equals the binomial transform of A008934 (number of tournament sequences): a(n) = Sum_{k=0..n} C(n, k)*A008934(k). - Paul D. Hanna, Sep 18 2005
|
|
|
MATHEMATICA
|
T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
a[n_] := Sum[T[n, k], {k, 0, n}];
|
|
|
PROG
|
(SageMath)
@CachedFunction
if k<0 or k>n: return 0
elif k==0 or k==n: return 1
else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A016121(n): return sum(T(n, k) for k in range(n+1))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A136555
|
|
Square array, read by antidiagonals, where T(n,k) = binomial(2^k + n-1, k).
|
|
+10
16
|
|
|
|
1, 1, 1, 1, 2, 3, 1, 3, 6, 35, 1, 4, 10, 56, 1365, 1, 5, 15, 84, 1820, 169911, 1, 6, 21, 120, 2380, 201376, 67945521, 1, 7, 28, 165, 3060, 237336, 74974368, 89356415775, 1, 8, 36, 220, 3876, 278256, 82598880, 94525795200, 396861704798625, 1, 9, 45, 286, 4845, 324632, 90858768, 99949406400, 409663695276000, 6098989894499557055
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Let vector R_{n} equal row n of this array; then R_{n+1} = P * R_{n} for n>=0, where triangle P = A132625 such that row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. for row n: Sum_{i>=0} (1 + 2^i*x)^(n-1) * log(1 + 2^i*x)^i / i!.
For the square array:
For the number triangle:
t(n, k) = T(n-k, k) = binomial(2^k + n - k -1, k).
Sum_{k=0..n} t(n,k) = Sum_{k=0..n} T(n-k, k) = A136557(n). (End)
|
|
|
EXAMPLE
|
Square array begins:
1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, ... A136556;
1, 2, 6, 56, 1820, 201376, 74974368, 94525795200, ... A014070;
1, 3, 10, 84, 2380, 237336, 82598880, 99949406400, ... A136505;
1, 4, 15, 120, 3060, 278256, 90858768, 105637584000, ... A136506;
1, 5, 21, 165, 3876, 324632, 99795696, 111600996000, ... ;
1, 6, 28, 220, 4845, 376992, 109453344, 117850651776, ... ;
1, 7, 36, 286, 5985, 435897, 119877472, 124397910208, ... ;
1, 8, 45, 364, 7315, 501942, 131115985, 131254487936, ... ;
...
Form column vector R_{n} out of row n of this array;
then row n+1 can be generated from row n by:
R_{n+1} = P * R_{n} for n>=0,
where triangular matrix P = A132625 begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
336, 60, 8, 1, 1;
25836, 2960, 248, 16, 1, 1;
6251504, 454072, 24800, 1008, 32, 1, 1; ...
where row n+1 of P = row n of P^(2^n) with appended '1' for n>=0.
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Binomial[2^k +n-k-1, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 14 2021 *)
|
|
|
PROG
|
(PARI) T(n, k)=binomial(2^k+n-1, k)
(PARI) /* Coefficient of x^k in g.f. of row n: */ T(n, k)=polcoeff(sum(i=0, k, (1+2^i*x+x*O(x^k))^(n-1)*log((1+2^i*x)+x*O(x^k))^i/i!), k)
(Sage) flatten([[binomial(2^k +n-k-1, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^k +n-k-1, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 14 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A163767
|
|
a(n) = tau_{n}(n) = number of ordered n-factorizations of n.
|
|
+10
15
|
|
|
|
1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(p) = p for prime p.
a(n) = n^k when n is the product of k distinct primes (conjecture).
a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
|
|
|
EXAMPLE
|
Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
The a(1) = 1 through a(5) = 5 chains of divisors:
() (1) (1/1) (1/1/1) (1/1/1/1)
(2) (3/1) (2/1/1) (5/1/1/1)
(3/3) (2/2/1) (5/5/1/1)
(2/2/2) (5/5/5/1)
(4/1/1) (5/5/5/5)
(4/2/1)
(4/2/2)
(4/4/1)
(4/4/2)
(4/4/4)
(End)
|
|
|
MATHEMATICA
|
Table[Times@@(Binomial[#+n-1, n-1]&/@FactorInteger[n][[All, 2]]), {n, 1, 50}] (* Enrique Pérez Herrero, Dec 25 2013 *)
|
|
|
PROG
|
(PARI) {a(n, m=n)=if(n==1, 1, if(m==1, 1, sumdiv(n, d, a(d, 1)*a(n/d, m-1))))}
(Python)
from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1, e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024
|
|
|
CROSSREFS
|
Diagonal n = k + 1 of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts nonempty strict chains of divisors of n.
A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
A343662 counts strict length-k chains of divisors (row sums: A337256).
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A136505
|
|
a(n) = binomial(2^n + 1, n).
|
|
+10
14
|
|
|
|
1, 3, 10, 84, 2380, 237336, 82598880, 99949406400, 422825581068000, 6318976181520699840, 337559127276933693852160, 65182103393445184131620004864, 45946437874792132748338425828443136
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: A(x) = Sum_{n>=0} (1 + 2^n*x) * log(1 + 2^n*x)^n/n!.
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) {a(n)=polcoeff(sum(i=0, n, (1+2^i*x +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!), n)}
(Sage) [binomial(2^n +1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
|
|
|
CROSSREFS
|
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), this sequence (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A136506
|
|
a(n) = binomial(2^n + 2, n).
|
|
+10
14
|
|
|
|
1, 4, 15, 120, 3060, 278256, 90858768, 105637584000, 436355999662176, 6431591598617108352, 340881559632021623909760, 65533747894341651530074060800, 46081376018330435634530315478453248
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: A(x) = Sum_{n>=0} (1 + 2^n*x)^2 * log(1 + 2^n*x)^n/n!.
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
Table[Binomial[2^n+2, n], {n, 0, 20}] (* Harvey P. Dale, Jun 20 2011 *)
|
|
|
PROG
|
(PARI) {a(n)=polcoeff(sum(i=0, n, (1+2^i*x +x*O(x^n))^2*log(1+2^i*x +x*O(x^n))^i/i!), n)}
(Sage) [binomial(2^n +2, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +2, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
|
|
|
CROSSREFS
|
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), this sequence (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A094223
|
|
Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).
|
|
+10
13
|
|
|
|
1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*binomial(2^k, n).
a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k+n-1, n).
|
|
|
MATHEMATICA
|
a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)
|
|
|
PROG
|
(PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022
|
|
|
CROSSREFS
|
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A132683
|
|
a(n) = binomial(2^n + n, n).
|
|
+10
13
|
|
|
|
1, 3, 15, 165, 4845, 435897, 131115985, 138432467745, 525783425977953, 7271150092378906305, 368539102493388126164865, 68777035446753808820521420545, 47450879627176629761462147774626305
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = [x^n] 1/(1-x)^(2^n + 1).
G.f.: Sum_{n>=0} (-log(1 - 2^n*x))^n / ((1 - 2^n*x)*n!). - Paul D. Hanna, Feb 25 2009
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 3*x + 15*x^2 + 165*x^3 + 4845*x^4 + 435897*x^5 + ...
A(x) = 1/(1-x) - log(1-2x)/(1-2x) + log(1-4x)^2/((1-4x)*2!) - log(1-8x)^3/((1-8x)*3!) +- ... (End)
|
|
|
MAPLE
|
|
|
|
MATHEMATICA
|
|
|
|
PROG
|
(PARI) a(n)=binomial(2^n+n, n)
(PARI) {a(n)=polcoeff(sum(m=0, n, (-log(1-2^m*x))^m/((1-2^m*x +x*O(x^n))*m!)), n)} \\ Paul D. Hanna, Feb 25 2009
(Sage) [binomial(2^n +n, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021
(Magma) [Binomial(2^n +n, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021
|
|
|
CROSSREFS
|
Sequences of the form binomial(2^n +p*n +q, n): A136556 (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), this sequence (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.023 seconds
|
