A213352 -id:A213352 - cp's OEIS frontend

A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= floor((n-1)/2).

1, 4, 12, 3, 32, 24, 80, 120, 10, 192, 480, 120, 448, 1680, 840, 35, 1024, 5376, 4480, 560, 2304, 16128, 20160, 5040, 126, 5120, 46080, 80640, 33600, 2520, 11264, 126720, 295680, 184800, 27720, 462, 24576, 337920, 1013760, 887040, 221760, 11088

Offset: 1

Stanislav Sykora, Jun 09 2012

[General discussion]: Consider the 2^N numbers with N-digit binary expansion. Let a pair (v,w), here called a "transition", be such that there are exactly k+q digits which are '0' in v and '1' in w, and exactly k digits which are '1' in v and '0' in w. Then T(q;N,k) is the number of all such pairs.
For given N and q, the rows of the triangle T(q;N,k) sum up to Sum[k]T(q;N,k) = C(2N,N-q) which is the total number of q-quantum transitions or, equivalently, the number of pairs in which the sum of binary digits of w exceeds that of v by exactly q (see Crossrefs).
The terminology stems from the mapping of the i-th digit onto quantum states of the i-th particle (-1/2 for digit '0', +1/2 for digit '1'), the numbers onto quantum states of the system, and the pairs onto quantum transitions between states. In magnetic resonance (NMR) the most intense transitions are the single-quantum ones (q=1) with k=0, called "main transitions", while those with k>0, called "combination transitions", tend to be weaker. Zero-, double- and, in general, q-quantum transitions are detectable by special techniques.
[Specific case]: This sequence is for single-quantum transitions (q = 1). It lists the flattened triangle T(1;N,k), with rows N = 1,2,... and columns k = 0..floor((N-1)/2).

			T(1;3,1) = 3 because the only transitions compatible with q=1,k=1 are (001,110),(010,101),(100,011).
Starting rows of the triangle T(1;N,k):
  N | k = 0, 1, ..., floor((N-1)/2)
  1 |  1
  2 |  4
  3 | 12   3
  4 | 32  24
  5 | 80 120 10

R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of nuclear magnetic resonance in one and two dimensions, Clarendon Press, 1987, Chapters 2-6.
M. H. Levitt, Spin Dynamics, J.Wiley & Sons, 2nd Ed.2007, Part3 (Section 6).
J. A. Pople, W. G. Schneider, H. J. Bernstein, High-resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Stanislav Sykora, Table of n, a(n) for n = 1..2550
Stanislav Sykora, T(1;N,k) with rows N=1,..,100 and columns k=0,..,floor((N-1)/2)
S. Sykora, p-Quantum Transitions and a Combinatorial Identity, Stan's Library, II, Aug 2007.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213344..A213352 (q=2..10).

Cf. A001787 (first column), A001791 (row sums).

egf := exp(2*x*y) * BesselI(1, 2*x):
ser := series(egf, x, 32): cx := n -> coeff(ser, x, n):
Trow := n -> n!*seq(coeff(cx(n), y, n - 2*k - 1), k = 0..floor((n-1)/2)):
seq(print([n], Trow(n)), n = 1..12); # Peter Luschny, May 12 2021

With[{q = 1}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 11}, {k, 0, Floor[(n - 1)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

TNQK(N, q, k)={binomial(N, k)*binomial(N-k, q+k)*2^((N-k)-(q+k))}
TQ(Nmax, q)={vector(Nmax-q+1, n, vector(1+(n-1)\2, k, TNQK(n+q-1, q, k-1)))}
{ concat(TQ(13, 1)) } \\ simplified by Andrew Howroyd, May 12 2021

Set q = 1 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

T(n, k) = n! * [y^(n-2*k-1)] [x^n] exp(2*x*y)*BesselI(1, 2*x). - Peter Luschny, May 12 2021

A213344 2-quantum transitions in systems of N>=2 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 6, 24, 4, 80, 40, 240, 240, 15, 672, 1120, 210, 1792, 4480, 1680, 56, 4608, 16128, 10080, 1008, 11520, 53760, 50400, 10080, 210, 28160, 168960, 221760, 73920, 4620, 67584, 506880, 887040, 443520, 55440, 792

Offset: 2

Stanislav Sykora, Jun 09 2012

For a general discussion, please see A213343.
This a(n) is for double-quantum transitions (q = 2).
It lists the flattened triangle T(2;N,k) with rows N = 2,3,... and columns N, k = 0..floor((N-2)/2).

			For N=4, there are 4 second-quantum transitions with combination index 1: (0001,1110),(0010,1101),(0100,1011),(1000,0111).
Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-2)/2)
  2 |   1
  3 |   6
  4 |  24   4
  5 |  80  40
  6 | 240 240 15

See A213343.

Stanislav Sykora, Table of n, a(n) for n = 2..2501
Stanislav Sykora, T(2;N,k) with rows N=2,..,100 and columns k=0,..,floor((N-2)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213345 to A213352 (q=3..10).

Cf. A001788 (first column), A002694 (row sums).

With[{q = 2}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 12}, {k, 0, Floor[(n - 2)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 2.
```

Set q = 2 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A213345 3-quantum transitions in systems of N>=3 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 8, 40, 5, 160, 60, 560, 420, 21, 1792, 2240, 336, 5376, 10080, 3024, 84, 15360, 40320, 20160, 1680, 42240, 147840, 110880, 18480, 330, 112640, 506880, 532224, 147840, 7920, 292864, 1647360, 2306304, 960960, 102960

Offset: 3

Stanislav Sykora, Jun 12 2012

For a general discussion, please see A213343.
This a(n) is for triple-quantum transitions (q = 3).
It lists the flattened triangle T(3;N,k) with rows N = 3,5,... and columns k = 0..floor((N-3)/2).

			Some of the 40 triple-quantum transitions for N = 5 and combination index 0: (00000,01011),(10010,11111),...
Starting rows of the triangle T(3;N,k):
  N | k = 0, 1, ..., floor((N-3)/2)
  3 |   1
  4 |   8
  5 |  40   5
  6 | 160  60
  7 | 560 420 21

See A213343

Stanislav Sykora, Table of n, a(n) for n = 3..2452
Stanislav Sykora, T(3;N,k) with rows N=3,..,100 and columns k=0,..,floor((N-3)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 (q=1), A213344 (q=2), A213346 to A213352 (q=4..10).

Cf. A001789 (first column), A002696 (row sums).

With[{q = 3}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 13}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 3.
```

Set q = 3 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A213346 4-quantum transitions in systems of N>=4 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 10, 60, 6, 280, 84, 1120, 672, 28, 4032, 4032, 504, 13440, 20160, 5040, 120, 42240, 88704, 36960, 2640, 126720, 354816, 221760, 31680, 495, 366080, 1317888, 1153152, 274560, 12870, 1025024, 4612608, 5381376, 1921920, 180180, 2002

Offset: 4

Stanislav Sykora, Jun 12 2012

For a general discussion, please see A213343.
This a(n) is for quadruple-quantum transitions (q = 4).
It lists the flattened triangle T(4;N,k) with rows N = 4,5,... and columns k = 0..floor((N-4)/2).

			Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-4)/2)
  4 |    1
  5 |   10
  6 |   60   6
  7 |  280  84
  8 | 1120 672 28

See A213343.

Stanislav Sykora, Table of n, a(n) for n = 4..2404
Stanislav Sykora, T(4;N,k) with rows N=4,..,100 and columns k=0,..,floor((N-4)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213345 (q=1 to 3), A213347 to A213352 (q=5 to 10).

Cf. A003472 (first column), A004310 (row sums).

With[{q = 4}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 14}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 4
```

Set q = 4 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k)

A213347 5-quantum transitions in systems of N>=5 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 12, 84, 7, 448, 112, 2016, 1008, 36, 8064, 6720, 720, 29568, 36960, 7920, 165, 101376, 177408, 63360, 3960, 329472, 768768, 411840, 51480, 715, 1025024, 3075072, 2306304, 480480, 20020, 3075072, 11531520, 11531520

Offset: 5

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for quintuple-quantum transitions (q = 5).
It lists the flattened triangle T(5;N,k) with rows N = 5,6,... and columns N, k = 0..floor((N-5)/2).

			Starting rows of the triangle:
  N | k = 0, 1, ..., floor((N-5)/2)
  5 |    1
  6 |   12
  7 |   84    7
  8 |  448  112
  9 | 2016 1008 36

See A213343

Stanislav Sykora, Table of n, a(n) for n = 5..2356
Stanislav Sykora, T(5;N,k) with rows N=5,..,100 and columns k=0,..,floor((N-5)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213346 (q=1 to 4), A213348 to A213352 (q=6 to 10).

A054849 (first column), A004311 (row sums).

With[{q = 5}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, 15}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 18 2019 *)

PARI
```
See A213343; set thisq = 5
```

Set q = 5 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A213348 6-quantum transitions in systems of N >= 6 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 14, 112, 8, 672, 144, 3360, 1440, 45, 14784, 10560, 990, 59136, 63360, 11880, 220, 219648, 329472, 102960, 5720, 768768, 1537536, 720720, 80080, 1001, 2562560, 6589440, 4324320, 800800, 30030, 8200192, 26357760, 23063040

Offset: 6

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for sextuple-quantum transitions (q = 6).
It lists the flattened triangle T(6;N,k) with rows N = 6,7,... and columns k = 0..floor((N-6)/2).

			Starting rows of the triangle:
   N | k = 0, 1, ..., floor((N-6)/2)
   6 |    1
   7 |   14
   8 |  112    8
   9 |  672  144
  10 | 3360 1440 45

See A213343

Stanislav Sykora, Table of n, a(n) for n = 6..2309
Stanislav Sykora, T(6;N,k) with rows N = 6..100 and columns k = 0..floor((N-6)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213347 (q=1 to 5), A213349 to A213352 (q=7 to 10).

Cf. A002409 (first column, with offset 6), A004312 (row sums).

With[{q = 6}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, q, q + 10}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 20 2019 *)

PARI
```
See A213343; set thisq = 6
```

Set q = 6 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A213349 7-quantum transitions in systems of N >= 7 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 16, 144, 9, 960, 180, 5280, 1980, 55, 25344, 15840, 1320, 109824, 102960, 17160, 286, 439296, 576576, 160160, 8008, 1647360, 2882880, 1201200, 120120, 1365, 5857280, 13178880, 7687680, 1281280, 43680, 19914752, 56010240

Offset: 7

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for septuple-quantum transitions (q = 7).
It lists the flattened triangle T(7;N,k) with rows N = 7,8,... and columns k = 0..floor((N-7)/2).

			Starting rows of the triangle:
   N | k = 0, 1, ..., floor((N-7)/2)
   7 |    1
   8 |   16
   9 |  144    9
  10 |  960  180
  11 | 5280 1980 55

See A213343

Stanislav Sykora, Table of n, a(n) for n = 7..2262
Stanislav Sykora, T(7;N,k) with rows N = 7..100 and columns k = 0..floor((N-7)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213348 (q=1 to 6), A213350 to A213352 (q=8 to 10).

Cf. A054851 (first column), A004313 (row sums).

With[{q = 7}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, q, q + 10}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 20 2019 *)

PARI
```
See A213343; set thisq = 7
```

Set q = 7 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

A213350 8-quantum transitions in systems of N >= 8 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 18, 180, 10, 1320, 220, 7920, 2640, 66, 41184, 22880, 1716, 192192, 160160, 24024, 364, 823680, 960960, 240240, 10920, 3294720, 5125120, 1921920, 174720, 1820, 12446720, 24893440, 13069056, 1980160, 61880, 44808192, 112020480

Offset: 8

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for octuple-quantum transitions (q = 8).
It lists the flattened triangle T(8;N,k) with rows N = 8,9,... and columns k = 0..floor((N-8)/2).

			Starting rows of the triangle:
   N | k = 0, 1, ..., floor((N-8)/2)
  ---+------------------------------
   8 |    1
   9 |   18
  10 |  180   10
  11 | 1320  220
  12 | 7920 2640 66

See A213343

Stanislav Sykora, Table of n, a(n) for n = 8..2216
Stanislav Sykora, T(8;N,k) with rows N = 8..100 and columns k = 0..floor((N-8)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213349 (q=1 to 7), A213351 (q=9), A213352 (q= 10).

Cf. A140325 (first row, with offset 8), A004314 (row sums).

With[{q = 8}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, q, q + 10}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 20 2019 *)

PARI
```
See A213343; set thisq = 8
```

Set q = 8 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k)

A213351 9-quantum transitions in systems of N >= 9 spin 1/2 particles, in columns by combination indices.

Original entry on oeis.org

1, 20, 220, 11, 1760, 264, 11440, 3432, 78, 64064, 32032, 2184, 320320, 240240, 32760, 455, 1464320, 1537536, 349440, 14560, 6223360, 8712704, 2970240, 247520, 2380, 24893440, 44808192, 21385728, 2970240, 85680, 94595072, 212838912, 135442944, 28217280

Offset: 9

Stanislav Sykora, Jun 13 2012

For a general discussion, please see A213343.
This a(n) is for nonuple-quantum transitions (q = 9).
It lists the flattened triangle T(9;N,k) with rows N = 9,10,... and columns k = floor((N-9)/2).

			Starting rows of the triangle:
   N | k = 0, 1, ..., floor((N-9)/2)
  ---+------------------------------
   9 | 1
  10 | 20
  11 | 220 11
  12 | 1760 264
  13 | 11440 3432 78

See A213343

Stanislav Sykora, Table of n, a(n) for n = 9..2170
Stanislav Sykora, T(9;N,k) with rows N = 9..100 and columns k = 0..floor((N-9)/2)
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.

Cf. A051288 (q=0), A213343 to A213350 (q=1 to 8), A213352 (q= 10).

Cf. A140354 (first column,with offset 9), A004315 (row sums).

With[{q = 9}, Table[2^(n - q - 2 k)*Binomial[n, k] Binomial[n - k, q + k], {n, q, q + 10}, {k, 0, Floor[(n - q)/2]}]] // Flatten (* Michael De Vlieger, Nov 20 2019 *)

PARI
```
\\ See A213343; set thisq = 9
```

Set q = 9 in: T(q;N,k) = 2^(N-q-2*k)*binomial(N,k)*binomial(N-k,q+k).

cp's OEIS Frontend

A213343 1-quantum transitions in systems of N spin 1/2 particles, in columns by combination indices. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= floor((n-1)/2).

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A213344 2-quantum transitions in systems of N>=2 spin 1/2 particles, in columns by combination indices.

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A213345 3-quantum transitions in systems of N>=3 spin 1/2 particles, in columns by combination indices.

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A213346 4-quantum transitions in systems of N>=4 spin 1/2 particles, in columns by combination indices.

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A213347 5-quantum transitions in systems of N>=5 spin 1/2 particles, in columns by combination indices.

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A213348 6-quantum transitions in systems of N >= 6 spin 1/2 particles, in columns by combination indices.

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