A247608 -id:A247608 - cp's OEIS frontend

A247612 a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).

1, 15, 120, 680, 3060, 11628, 38760, 116280, 316767, 788161, 1805100, 3840420, 7660250, 14446134, 25947612, 44668692, 74091645, 118941555, 185495056, 281936688, 418766304, 609260960, 869994720, 1221419808, 1688512539, 2301487461, 3096583140, 4116923020

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A005408, A056108, A247608 - A247611

[(1680+386012*n-958048*n^2+943761*n^3-455455*n^4+123123*n^5- 17017*n^6+1144*n^7)/1680: n in [0..40]];

I:=[1,15,120,680,3060,11628,38760, 116280]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]];

Table[(1680 + 386012 n - 958048 n^2 + 943761 n^3 - 455455 n^4 + 123123 n^5 - 17017 n^6 + 1144 n^7)/1680, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 7 x + 28 x^2 + 84 x^3 + 210 x^4 + 462 x^5 + 924 x^6 + 1716 x^7)/(1 - x)^8, {x, 0, 40}], x]
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,15,120,680,3060,11628,38760,116280},30] (* Harvey P. Dale, May 12 2017 *)

m=7; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 7*x + 28*x^2 + 84*x^3 + 210*x^4 + 462*x^5 + 924*x^6 + 1716*x^7) / (1-x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) = (1680 + 386012*n - 958048*n^2 + 943761*n^3 - 455455*n^4 + 123123*n^5 - 17017*n^6 + 1144*n^7)/1680.

Definition edited by Robert Israel, Sep 22 2014

A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

Original entry on oeis.org

1, 9, 45, 165, 495, 1231, 2639, 5055, 8885, 14605, 22761, 33969, 48915, 68355, 93115, 124091, 162249, 208625, 264325, 330525, 408471, 499479, 604935, 726295, 865085, 1022901, 1201409, 1402345, 1627515, 1878795, 2158131, 2467539, 2809105, 3184985, 3597405

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005408, A056108, A247608.

[(12-58*n+217*n^2-98*n^3+35*n^4)/12: n in [0..40]];

[1+8*Binomial(n, 1)+28*Binomial(n, 2)+56*Binomial(n, 3)+70*Binomial(n,4): n in [0..40]];

I:=[1,9,45,165,495]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Table[(12 - 58 n + 217 n^2 - 98 n^3 + 35 n^4)/12, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 4 x + 10 x^2 + 20 x^3 + 35 x^4)/(1 - x)^5, {x, 0, 50}], x]
LinearRecurrence[{5,-10,10,-5,1},{1,9,45,165,495},40] (* Harvey P. Dale, Oct 19 2024 *)

m=4; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (12 - 58*n + 217*n^2 - 98*n^3 + 35*n^4)/12.
a(n) = 1 + 8*Binomial(n, 1) + 28*Binomial(n, 2) + 56*Binomial(n, 3) + 70*Binomial(n, 4).

A247611 a(n) = Sum_{k=0..6} binomial(12,k)*binomial(n,k).

Original entry on oeis.org

1, 13, 91, 455, 1820, 6188, 18564, 49596, 119139, 260743, 527065, 996205, 1778966, 3027038, 4942106, 7785882, 11891061, 17673201, 25643527, 36422659, 50755264, 69525632, 93774176, 124714856, 163753527, 212507211, 272824293, 346805641, 436826650

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A005408, A056108, A247608, A247609.

[(120-8042*n+20581*n^2-17380*n^3+7645*n^4-1518*n^5+ 154*n^6)/120: n in [0..40]];

I:=[1,13,91,455,1820,6188,18564]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]];

Table[(120 - 8042 n + 20581 n^2 - 17380 n^3 + 7645 n^4 - 1518 n^5 + 154 n^6)/120, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 6 x + 21 x^2 + 56 x^3 + 126 x^4 + 252 x^5 + 462 x^6)/(1 - x)^7, {x, 0, 40}], x]

m=6; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 6*x + 21*x^2 + 56*x^3 + 126*x^4 + 252*x^5 + 462*x^6) / (1-x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
a(n) = (120 - 8042*n + 20581*n^2 - 17380*n^3 + 7645*n^4 -1518*n^5 + 154*n^6)/120.

A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).

Original entry on oeis.org

1, 11, 66, 286, 1001, 3003, 7798, 17858, 36873, 70003, 124130, 208110, 333025, 512435, 762630, 1102882, 1555697, 2147067, 2906722, 3868382, 5070009, 6554059, 8367734, 10563234, 13198009, 16335011, 20042946, 24396526, 29476721, 35371011, 42173638, 49985858

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A005408, A056108, A247608.

[(20+508*n-925*n^2+820*n^3-245*n^4+42*n^5)/20: n in [0..40]];

I:=[1,11,66,286,1001,3003]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]];

Table[(20 + 508 n - 925 n^2 + 820 n^3 - 245 n^4 + 42 n^5)/20, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 5 x + 15 x^2 + 35 x^3 + 70 x^4 + 126 x^5)/(1 - x)^6, {x, 0, 40}], x]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,66,286,1001,3003},40] (* Harvey P. Dale, Apr 20 2022 *)

m=5; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 126*x^5) / (1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (20 + 508*n - 925*n^2 + 820*n^3 - 245*n^4 + 42*n^5)/20.

A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 69, 56, 28, 8, 1, 1, 9, 36, 84, 121, 121, 84, 36, 9, 1, 1, 10, 45, 120, 195, 226, 195, 120, 45, 10, 1

Offset: 0

Kolosov Petro, Jul 08 2024

Triangle T(n,k) is the third triangle R3 among the rascal-family triangles; A077028 is triangle R1, A374378 is triangle R2.
Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 7).
Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 3).

			Triangle begins:
--------------------------------------------------
k=     0   1   2   3    4    5    6   7   8   9 10
--------------------------------------------------
n=0:   1
n=1:   1   1
n=2:   1   2   1
n=3:   1   3   3   1
n=4:   1   4   6   4    1
n=5:   1   5  10  10    5    1
n=6:   1   6  15  20   15    6    1
n=7:   1   7  21  35   35   21    7   1
n=8:   1   8  28  56   69   56   28   8   1
n=9:   1   9  36  84  121  121   84  36   9   1
n=10:  1  10  45 120  195  226  195  120  45  10  1

Jena Gregory, Brandt Kronholm and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.

Cf. A077028, A000027, A000217, A000292, A005894, A247608, A008860, A374378, A007318, A096943, A096940

t[n_, k_] := Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 3}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Left]

T(n,k) = 1 + k*(n-k) + 1/4*(k-1)*k*(n-k-1)*(n-k) + 1/36*(k-2)*(k-1)*k*(n-k-2)*(n-k-1)*(n-k).
Row sums give A008860(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A000292(n).
Diagonal T(n+4, n) gives A005894(n).
Diagonal T(n+6, n) gives A247608(n).
Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).
Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.
G.f.: (1 + 4*x^6*y^3 - 3*x*(1 + y) - 6*x^5*y^2*(1 + y) + 2*x^4*y*(2 + 7*y+ 2*y^2) + x^2*(3 + 10*y + 3*y^2) - x^3*(1 + 11*y + 11*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 09 2024

A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

Original entry on oeis.org

1, 17, 153, 969, 4845, 20349, 74613, 245157, 735471, 2031535, 5189327, 12316239, 27322191, 57029103, 112740255, 212383935, 383358645, 666220005, 1119362365, 1824861005, 2895653673, 4484253081, 6793194849, 10087438257, 14708950035, 21093714291

Offset: 0

Vincenzo Librandi, Sep 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A005408, A056108, A247608 - A247612.

m:=8; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];

[(20160-15076944*n+40499716*n^2-42247940*n^3 +23174515*n^4-7234136*n^5+1335334*n^6-134420*n^7 +6435*n^8)/20160: n in [0..40]];

Table[(20160 - 15076944 n + 40499716 n^2 - 42247940 n^3 + 23174515 n^4 - 7234136 n^5 + 1335334 n^6 - 134420 n^7 + 6435 n^8)/20160, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 8 x + 36 x^2 + 120 x^3 + 330 x^4 + 792 x^5 + 1716 x^6 + 3432 x^7 + 6435 x^8)/(1 - x)^9, {x, 0, 40}], x]
Table[Sum[Binomial[16,k]Binomial[n,k],{k,0,8}],{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,153,969,4845,20349,74613,245157,735471},40] (* Harvey P. Dale, Mar 25 2015 *)

m=8; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

G.f.: (1 + 8*x + 36*x^2 + 120*x^3 + 330*x^4 + 792*x^5 + 1716*x^6 + 3432*x^7  + 6435*x^8) / (1-x)^9.
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).
a(n) = (20160 - 15076944*n + 40499716*n^2 - 42247940*n^3 + 23174515*n^4 - 7234136*n^5 + 1335334*n^6 - 134420*n^7 + 6435*n^8) / 20160.

A247614 a(n) = Sum_{k=0..9} binomial(18,k)*binomial(n,k).

Original entry on oeis.org

1, 19, 190, 1330, 7315, 33649, 134596, 480700, 1562275, 4686825, 13079352, 34084128, 83204745, 191006115, 414237570, 852920310, 1675575165, 3155247975, 5719519850, 10018268150, 17013571223, 28096825757, 45238870040, 71179679480, 109665022415

Offset: 0

Vincenzo Librandi, Sep 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A005408, A056108, A247608 - A247612.

m:=9; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];

[(181440+462101904*n-1283316876*n^2+1433031524*n^3 -853620201*n^4+303063726*n^5-66245634*n^6 +8905416*n^7-678249*n^8+24310*n^9)/181440: n in [0..40]];

Table[(181440 + 462101904 n - 1283316876 n^2 + 1433031524 n^3 - 853620201 n^4 + 303063726 n^5 - 66245634 n^6 + 8905416 n^7 - 678249 n^8 + 24310 n^9)/181440, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 9 x + 45 x^2 + 165 x^3 + 495 x^4 + 1287 x^5 + 3003 x^6 + 6435 x^7 + 12870 x^8 + 24310 x^9)/(1 - x)^10, {x, 0, 40}], x]
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,19,190,1330,7315,33649,134596,480700,1562275,4686825},30] (* Harvey P. Dale, Jul 19 2019 *)

m=9; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

G.f.: (1 +9*x +45*x^2 +165*x^3 +495*x^4 +1287*x^5 +3003*x^6 + 6435*x^7 +12870*x^8 +24310*x^9) / (1-x)^10.
a(n) = 10*a(n-1) -45*a(n-2) +120*a(n-3) -210*a(n-4) +252*a(n-5) -210*a(n-6) +120*a(n-7) -45*a(n-8) +10*a(n-9) -a(n-10).
a(n) = (181440 + 462101904*n - 1283316876*n^2 + 1433031524*n^3 - 853620201*n^4 + 303063726*n^5 - 66245634*n^6+8905416*n^7 - 678249*n^8 + 24310*n^9) / 181440.

A247615 a(n) = Sum_{k=0..10} binomial(20,k)*binomial(n,k).

Original entry on oeis.org

1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015, 84504355, 223651350, 558350430, 1318250890, 2952624906, 6296642121, 12834146941, 25098124271, 47262174531, 85990654178, 151631858378, 259857912678, 433877085278, 707369215553

Offset: 0

Vincenzo Librandi, Sep 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A005408, A056108, A247608, A247609, A247610, A247611, A247612.

m:=10; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];

I:=[1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015]; [n le 11 select I[n] else 11*Self(n-1) -55*Self(n-2) +165*Self(n-3) -330*Self(n-4) +462*Self(n-5) -462*Self(n-6) +330*Self(n-7) -165*Self(n-8) +55*Self(n-9) -11*Self(n-10) +Self(n-11): n in [1..40]];

CoefficientList[Series[(1 + 10 x + 55 x^2 + 220 x^3 + 715 x^4 + 2002 x^5 + 5005 x^6 + 11440 x^7 + 24310 x^8 + 48620 x^9 + 92378 x^10)/(1 - x)^11, {x, 0, 40}], x]
Table[Sum[Binomial[20,k]Binomial[n,k],{k,0,10}],{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,21,231,1771,10626,53130,230230,888030,3108105,10015005,30045015},30] (* Harvey P. Dale, May 19 2015 *)

m=10; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 23 2014

G.f.: (1 + 10*x + 55*x^2 + 220*x^3 + 715*x^4 + 2002*x^5 + 5005*x^6 + 11440*x^7 + 24310*x^8 + 48620*x^9 + 92378*x^10) / (1-x)^11.
a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6)+330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +a(n-11).
a(n) = 1 - 5512999*n/630 + 212329883*n^2/8400 - 134689309*n^3/4536 + 3453077689*n^4/181440 - 64212077*n^5/8640 + 80300707*n^6/43200 - 1817521*n^7/6048 + 1860157*n^8/60480 - 331721*n^9/181440 + 46189*n^10/907200.

cp's OEIS Frontend

A247612 a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).

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A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

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A247611 a(n) = Sum_{k=0..6} binomial(12,k)*binomial(n,k).

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A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).

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A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

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A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

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A247614 a(n) = Sum_{k=0..9} binomial(18,k)*binomial(n,k).

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