A000581 -id:A000581 - cp's OEIS frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

A053310 a(n) = (n+3)*binomial(n+8, 8)/3.

Original entry on oeis.org

1, 12, 75, 330, 1155, 3432, 9009, 21450, 47190, 97240, 189618, 352716, 629850, 1085280, 1812030, 2941884, 4657983, 7210500, 10935925, 16280550, 23828805, 34337160, 48774375, 68368950, 94664700, 129585456, 175509972, 235358200

Offset: 0

Barry E. Williams, Mar 06 2000

If Y is a 3-subset of an n-set X then, for n>=11, a(n-11) is the number of 11-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Partial sums of A053367.
Cf. A053367, A053347, A000581.
Cf. A093560 ((3, 1) Pascal, column m=9).

[(n+3)*Binomial(n+8, 8)/3: n in [0..30]]; // G. C. Greubel, May 24 2018

CoefficientList[Series[(1+2*x)/(1-x)^10, {x, 0, 50}], x] (* G. C. Greubel, May 24 2018 *)
Table[(n+3) Binomial[n+8,8]/3,{n,0,30}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,12,75,330,1155,3432,9009,21450,47190,97240},30] (* Harvey P. Dale, Feb 25 2021 *)

for(n=0, 30, print1((n+3)*binomial(n+8, 8)/3, ", ")) \\ G. C. Greubel, May 24 2018

G.f.: (1+2*x)/(1-x)^10.

a(n) = binomial(n+8,n+2)*binomial(n+3,n)/28. - Zerinvary Lajos, May 12 2006

A095664 Ninth column (m=8) of (1,3)-Pascal triangle A095660.

Original entry on oeis.org

3, 25, 117, 405, 1155, 2871, 6435, 13299, 25740, 47190, 82654, 139230, 226746, 358530, 552330, 831402, 1225785, 1773783, 2523675, 3535675, 4884165, 6660225, 8974485, 11960325, 15777450, 20615868, 26700300, 34295052, 43709380, 55303380, 69494436, 86764260, 107666559, 132835365, 162994065

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 3-subset of an n-set X then, for n>=10, a(n-10) is the number of 8-subsets of X having at most one element in common with Y. - Milan Janjic, Nov 23 2007

Eighth column: A095663. Tenth column: A095665.

x='x+O('x^66); Vec((3-2*x)/(1-x)^9) \\ Joerg Arndt, May 11 2013

a(n)= binomial(n+7, 7)*(n+24)/8 = 3*b(n)-2*b(n-1), with b(n):=binomial(n+8, 8); cf. A000581.

G.f.: (3-2*x)/(1-x)^9.

A166813 Number of n X 8 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nondecreasing order.

Original entry on oeis.org

7, 43, 163, 493, 1285, 3001, 6433, 12868, 24308, 43756, 75580, 125968, 203488, 319768, 490312, 735469, 1081573, 1562273, 2220073, 3108103, 4292143, 5852923, 7888723, 10518298, 13884154, 18156202, 23535818, 30260338, 38608018, 48903490, 61523746, 76904683

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

			Some solutions for n=4
...1.1.1.1.1.2.2.2...1.1.1.1.1.1.2.2...1.1.1.1.1.1.2.2...1.1.1.1.1.1.2.2
...1.1.1.1.1.2.2.2...1.1.1.2.2.2.2.2...1.1.1.1.1.1.2.2...1.1.1.1.1.2.2.2
...1.2.2.2.2.2.2.2...1.1.1.2.2.2.2.2...1.1.1.2.2.2.2.2...1.1.1.2.2.2.2.2
...1.2.2.2.2.2.2.2...1.1.2.2.2.2.2.2...1.1.2.2.2.2.2.2...1.1.1.2.2.2.2.2
------
...1.1.1.1.1.1.1.1...1.1.1.1.1.1.1.2...1.1.1.1.1.1.1.2...1.1.1.1.1.1.2.2
...1.1.1.1.1.1.1.1...1.1.1.1.1.1.1.2...1.1.1.1.1.2.2.2...1.2.2.2.2.2.2.2
...1.2.2.2.2.2.2.2...1.1.1.1.1.1.1.2...1.1.1.2.2.2.2.2...2.2.2.2.2.2.2.2
...1.2.2.2.2.2.2.2...1.2.2.2.2.2.2.2...1.1.2.2.2.2.2.2...2.2.2.2.2.2.2.2

G. C. Greubel, Table of n, a(n) for n = 1..1000

a:= n-> binomial(n+8,8)-2:
seq (a(n), n=1..40);  # Alois P. Heinz, May 31 2012

Table[Binomial[n+8,8] -2, {n, 1, 100}] (* G. C. Greubel, May 24 2016 *)

a(n) = A000581(n+8)-2. - Alois P. Heinz, May 31 2012
From G. C. Greubel, May 24 2016: (Start)
G.f.: 1/(1-x)^9 - (1+x)/(1-x).
E.g.f.: (1/8!)*(-40320 + 322560*x + 564480*x^2 + 376320*x^3 + 117600*x^4 + 18816*x^5 + 1568*x^6 + 64*x^7 + x^8)*exp(x) + 1. (End)

A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 61, 66, 69, 70, 69, 66, 61, 54, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 279, 342, 405, 465, 519, 564, 597, 615, 615, 597, 564, 519, 465, 405, 342, 279, 219, 165, 120, 84

Offset: 1

Miquel Cerda, Jul 05 2017

The m-th row is palindromic; T(m,k) = T(m,9*m+1-k).

			The irregular triangle T(m,k) begins:
m\k  1  2  3  4  5   6   7   8   9   10   11  12   13   14  15  16  17  18  19
1    1  1  1  1  1   1   1   1   1;
2    1  2  3  4  5   6   7   8   9    9    8   7    6    5   4   3   2   1;
3    1  3  6  10 15  21  28  36  45   54   61  66   69   70  69  66  61  54 45,...;
4    1  4  10 20 35  56  84  120 165  219  279 342  405  465,...;
5    1  5  15 35 70  126 210 330 495  714  992 1330 1725,...;
6    1  6  21 56 126 252 462 792 1287 2001 2992,...;
etc.
Row m(2), column k(4) there are 4 numbers of 2-digits whose digits sum = 4: 13, 22, 31, 40.

Miquel Cerda, Rows n=1..10 of triangle, flattened

The row sums = 9*10^(m-1) = A052268(n). The row lengths = 9*m = A008591(n). The middle diagonal = A071976. (row m=3) = A071817, (row m=4) = A090579, (row m=5) = A090580, (row m=6) = A090581, (row m=7) = A278969, (row m=8) = A278971, (row m=9) = A289354, (column k=3) = A000217, (column k=4) = A000292, (column k=5) = A000332, (column k=6) = A000389, (column k=7) = A000579, (column k=8) = A000580, (column k=9) = A000581, (column k=10) = A035927.

row:= proc(m) local g; g:= normal((1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m);
seq(coeff(g,x,j),j=1..9*m) end proc:
seq(row(k),k=1..5); # Robert Israel, Jul 19 2017

G.f. of row m: (1 - x^10)^(m-1)*(x - x^10)/(1 - x)^m.

G.f. as array: (1+x+x^2)*(1+x^3+x^6)*x*y/(1-y*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)). - Robert Israel, Jul 19 2017

Edited by Robert Israel, Jul 19 2017

A096946 Ninth column of (1,5)-Pascal triangle A096940.

Original entry on oeis.org

5, 41, 189, 645, 1815, 4455, 9867, 20163, 38610, 70070, 121550, 202878, 327522, 513570, 784890, 1172490, 1716099, 2465991, 3485075, 4851275, 6660225, 9028305, 12096045, 16031925, 21036600, 27347580, 35244396, 45054284, 57158420

Offset: 0

Wolfdieter Lang, Jul 16 2004

If Y is a 5-subset of an n-set X then, for n>=12, a(n-12) is the number of 8-subsets of X having at most one element in common with Y. > - Milan Janjic, Dec 08 2007

Eighth column: A096945; tenth column: A096947.

a(n)= (n+40)*binomial(n+7, 7)/8 = 5*b(n)-4*b(n-1), with b(n):=A000581(n+8)=binomial(n+8, 8).

G.f.: (5-4*x)/(1-x)^9.

A104475 a(n) = binomial(n+4,4) * binomial(n+8,4).

Original entry on oeis.org

70, 630, 3150, 11550, 34650, 90090, 210210, 450450, 900900, 1701700, 3063060, 5290740, 8817900, 14244300, 22383900, 34321980, 51482970, 75710250, 109359250, 155405250, 217567350, 300450150, 409704750, 552210750, 736281000, 971890920, 1270934280, 1647507400, 2118223800

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0): C(0+4,4)*C(0+8,4) = C(4,4)*C(8,4) = 1*70 = 70.
a(7): C(5+4,4)*C(5+8,4) = C(9,4)*(13,4) = 126*715 = 90090.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000581, A062264.

[Binomial(n+4,4)*Binomial(n+8,4): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

A104475:=n->binomial(n+4,4)*binomial(n+8,4): seq(A104475(n), n=0..40); # Wesley Ivan Hurt, Jan 29 2017

f[n_] := Binomial[n + 4, 4]Binomial[n + 8, 4]; Table[ f[n], {n, 0, 25}] (* Robert G. Wilson v, Apr 20 2005 *)

vector(30, n, n--; binomial(n+4,4)*binomial(n+8,4)) \\ Michel Marcus, Jul 31 2015

def A104475(n): return binomial(n+4,4)*binomial(n+8,4)
print([A104475(n) for n in range(31)]) # G. C. Greubel, Mar 05 2025

a(n) = 70*A000581(n-8). - Michel Marcus, Jul 31 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/245.
Sum_{n>=0} (-1)^n/a(n) = 512*log(2)/35 - 37216/3675. (End)
From G. C. Greubel, Mar 05 2025: (Start)
G.f.: 70/(1-x)^9.
E.g.f.: (1/576)*(40320 + 322560*x + 564480*x^2 + 376320*x^3 + 117600*x^4 + 18816*x^5 + 1568*x^6 + 64*x^7 + x^8)*exp(x). (End)

More terms from Robert G. Wilson v, Apr 20 2005

A104478 a(n) = binomial(n+8,8)*binomial(n+12,8).

Original entry on oeis.org

495, 11583, 135135, 1061775, 6370650, 31286970, 131405274, 486370170, 1621233900, 4946841900, 13992495660, 37058912748, 92647281870, 220089696750, 499568676750, 1088533853550, 2285921092455, 4642276728375, 9143878404375, 17513561154375, 32691980821500, 59592810754620

Offset: 0

Zerinvary Lajos, Apr 18 2005

All terms are multiples of 99. - Michel Marcus, Aug 01 2015

			a(0): C(0+8,8)*C(0+12,8) = C(8,8)*C(12,8) = 1*495 = 495.
a(7): C(7+8,8)*C(7+12,8) = C(15,8)*C(19,8) = 6435*75582 = 486370170.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A000581, A062264.

[Binomial(n+8,8)*Binomial(n+12,8): n in [0..30]]; // Vincenzo Librandi, Jul 31 2015

f[n_] := Binomial[n + 8, 8] * Binomial[n + 12, 8]; Table[ f[n], {n, 0, 18}] (* Robert G. Wilson v, Apr 19 2005 *)

vector(30, n, n--; binomial(n+8,8)*binomial(n+12,8)) \\ Michel Marcus, Jul 31 2015

def A104478(n): return binomial(n+8,8)*binomial(n+12,8)
print([A104478(n) for n in range(31)]) # G. C. Greubel, Mar 04 2025

a(n) = A000581(n+8)*A000581(n+12). - Michel Marcus, Aug 01 2015
From Amiram Eldar, Sep 04 2022: (Start)
Sum_{n>=0} 1/a(n) = 11648*Pi^2/3 - 65726161036/1715175.
Sum_{n>=0} (-1)^n/a(n) = 262144*log(2)/99 - 629604992/343035. (End)
G.f.: 99*(5 + 32*x + 56*x^2 + 32*x^3 + 5*x^4)/(1-x)^17. - G. C. Greubel, Mar 04 2025

Corrected and extended by Robert G. Wilson v, Apr 19 2005

a(6) corrected by Georg Fischer, May 08 2021

A323538 a(n) = Product_{k=1..n} (binomial(k-1,8) + binomial(n-k,8)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 676788847291127500630565286687890625, 224202413239751513418389758669186941328125000, 81789054189516490351294844356948943677175390625000000, 29455964980491136378751203264203423123185624125549245000000000

Offset: 0

Vaclav Kotesovec, Jan 17 2019

nonn

Cf. A000581, A323425, A323496, A323497, A323533, A323534, A323535.

Table[Product[Binomial[k-1,8] + Binomial[n-k,8], {k, 1, n}], {n, 0, 20}]

a(n) = prod(k=1, n, binomial(k-1, 8)+binomial(n-k, 8)) \\ Felix Fröhlich, Jan 17 2019

a(n) ~ exp(-8*n + (2*sqrt(2+sqrt(2)) - 1/2 - sqrt(2))*Pi*(n-8)) * n^(8*n) / (8!)^n.

A095671 Ninth column (m=8) of (1,4)-Pascal triangle A095666.

Original entry on oeis.org

4, 33, 153, 525, 1485, 3663, 8151, 16731, 32175, 58630, 102102, 171054, 277134, 436050, 668610, 1001946, 1470942, 2119887, 3004375, 4193475, 5772195, 7844265, 10535265, 13996125, 18407025, 23981724, 30972348, 39674668, 50433900

Offset: 0

Wolfdieter Lang, Jun 11 2004

If Y is a 4-subset of an n-set X then, for n>=11, a(n-11) is the number of 8-subsets of X having at most one element in common with Y. - Milan Janjic, Dec 08 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

CoefficientList[Series[(4-3x)/(1-x)^9,{x,0,30}],x] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{4,33,153,525,1485,3663,8151,16731,32175},30] (* Harvey P. Dale, Jul 07 2015 *)

A095671(n):=(n+32)*binomial(n+7, 7)/8$
makelist(A095671(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

G.f.: (4-3*x)/(1-x)^9.
a(n) = 4*b(n)-3*b(n-1) =(n+32)*binomial(n+7, 7)/8, with b(n):=binomial(n+8, 8)=A000581(n+8, 8).
a(0)=4, a(1)=33, a(2)=153, a(3)=525, a(4)=1485, a(5)=3663, a(6)=8151, a(7)=16731, a(8)=32175, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ 126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Jul 07 2015

cp's OEIS Frontend

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

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A053310 a(n) = (n+3)*binomial(n+8, 8)/3.

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A095664 Ninth column (m=8) of (1,3)-Pascal triangle A095660.

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A166813 Number of n X 8 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nondecreasing order.

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A289410 Irregular triangular array T(m,k) with m (row) >= 1 and k (column) >= 1 read by rows: number of m-digit numbers whose digit sum is k.

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A096946 Ninth column of (1,5)-Pascal triangle A096940.

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A104475 a(n) = binomial(n+4,4) * binomial(n+8,4).

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A104478 a(n) = binomial(n+8,8)*binomial(n+12,8).

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