A017761 -id:A017761 - cp's OEIS frontend

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A017760 Binomial coefficients C(n,96).

Original entry on oeis.org

1, 97, 4753, 156849, 3921225, 79208745, 1346548665, 19813501785, 257575523205, 3005047770725, 31853506369685, 309847743777845, 2788629694000605, 23381587434312765, 183712472698171725, 1359472297966470765, 9516306085765295355, 63255446334792845595

Offset: 96

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 96..10000

Cf. A017761, A017762, A001787, A242091.

[Binomial(n,96): n in [96..110]]; // G. C. Greubel, Nov 12 2018

Table[Binomial[n, 96], {n, 96, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a(n)=binomial(n,96) \\ Charles R Greathouse IV, Jun 28 2012

[binomial(n, 96) for n in range(96,111)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^96/(1-x)^97.
E.g.f.: x^96*exp(x)/96!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=96} 1/a(n) = 96/95.
Sum_{n>=96} (-1)^n/a(n) = A001787(96)*log(2) - A242091(96)/95! = 3802951800684688204490109616128*log(2) - 9868088483131918614290277915496170231743780878869426700864981897216 / 3743576848674424246377459672213933825 = 0.9898949826... (End)

A017758 Binomial coefficients C(n,94).

Original entry on oeis.org

1, 95, 4560, 147440, 3612280, 71523144, 1192052400, 17199613200, 219295068300, 2509710226100, 26100986351440, 249145778809200, 2200787712814600, 18114175790089400, 139737927523546800, 1015428940004440080, 6981073962530525550, 45582306461228725650

Offset: 94

N. J. A. Sloane

nonn

Michael De Vlieger, Table of n, a(n) for n = 94..10000

Cf. A017760, A017761, A001787, A242091.

[Binomial(n,94): n in [94..110]]; // G. C. Greubel, Nov 12 2018

Table[Binomial[n, 94], {n, 94, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

for(n=94, 110, print1(binomial(n,94), ", ")) \\ G. C. Greubel, Nov 12 2018

[binomial(n, 94) for n in range(93,110)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^94/(1-x)^95.
E.g.f.: x^94*exp(x)/94!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=94} 1/a(n) = 94/93.
Sum_{n>=94} (-1)^n/a(n) = A001787(94)*log(2) - A242091(94)/93! = 930930909542605966724141416448*log(2) - 154188882548936228348285592429449074905987023631107827159541306779 / 238951713745176015726220830141314925 = 0.9896864636... (End)

A017759 Binomial coefficients C(n,95).

Original entry on oeis.org

1, 96, 4656, 152096, 3764376, 75287520, 1267339920, 18466953120, 237762021420, 2747472247520, 28848458598960, 277994237408160, 2478781950222760, 20592957740312160, 160330885263858960, 1175759825268299040, 8156833787798824590, 53739140249027550240

Offset: 95

N. J. A. Sloane

Michael De Vlieger, Table of n, a(n) for n = 95..10000

Cf. A017760, A017761, A001787, A242091.

[Binomial(n,95): n in [95..110]]; // G. C. Greubel, Nov 12 2018

Table[Binomial[n, 95], {n, 95, 120}] (* Vladimir Joseph Stephan Orlovsky, Jul 15 2011 *)

a(n)=binomial(n,95) \\ Charles R Greathouse IV, Jun 28 2012

[binomial(n, 95) for n in range(95,110)] # Zerinvary Lajos, May 23 2009

From G. C. Greubel, Nov 12 2018: (Start)
G.f.: x^95/(1-x)^96.
E.g.f.: x^95*exp(x)/95!. (End)
From Amiram Eldar, Dec 20 2020: (Start)
Sum_{n>=95} 1/a(n) = 95/94.
Sum_{n>=95} (-1)^(n+1)/a(n) = A001787(95)*log(2) - A242091(95)/94! = 1881668859713778017846668820480*log(2) - 308377765097872456696571184859137101525719223277941875149223928483 / 236436432547858373455418505613511610 = 0.9897917882... (End)

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A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

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A017760 Binomial coefficients C(n,96).

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A017758 Binomial coefficients C(n,94).

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A017759 Binomial coefficients C(n,95).

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