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binomial(i,j)*11^(i-j)*7^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:             576,              350,               49;\n  1:           12636,            12096,             3675,              343;\n  2:          268272,           353808,           169344,            34300,             2401;\n  3:         5563728,          9389520,          6191640,          1975680,           300125,           16807;\n  4:       113374080,        233676576,        197179920,         86682960,         20744640,         2521050,          117649;\n  5:      2278819008,       5555329920,       5725076112,       3220605360,       1061866260,       203297472,        20588575,         823543;\n  6:     45304282368,     127613864448,     155549237760,     106868087424,      45088475040,     11892902112,      1897443072,      164708600,        5764801;\n  7:    892616806656,    2854169789184,    4019836730112,    3266533992960,    1683172376928,    568114785504,    124875472176,    17076987648,     1297080225,      40353607;\n  8:  17455617552384,   62483176465920,   99895942621440,   93796190369280,   57164344876800,  23564413276992,   6628005830880,  1248754721760,   149423641920,   10088401750,   282475249;\n  9: 339194386529280, 1344082551533568, 2405602293937920, 2563995860616960, 1805576664608640, 880330911102720, 302409970388064, 72908064139680, 12019264196940, 1278402269760, 77680693475, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy (7*n+25+7*x)*P(n,x) = (126*n+576+(49*n+350)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 350*x + 49*x^2.","mathematica":"P[0, x_] := 576 + 350*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((126*n + 576 + (49*n + 350)*x + 49*x^2)*P[n - 1, x])/(7*n + 25 + 7*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 576 + 350*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 576 + (49*n + 350)*x + 49*x^2)*v[i - 1])/(7*n + 25 + 7*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A007318","D+1"],"edges":["e-5a47242ff4b92549","e-12d1d35f71c2af5c"],"text":"A038321 \u2192 basis A007318 \u2192 D+1"},"headline":"A038321 \u2192 basis A007318 \u2192 D+1","chain":null}