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T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","3"],["30","42","9"],["210","510","243","27"],["1680","6240","4932","1188","81"],["15120","79920","92700","36180","5265","243"],["151200","1084320","1722960","965520","224370","21870","729"],["1663200","15634080","32543280","24479280","8057070","1241730","86751","2187"],["19958400","239500800","632681280","611694720","266807520","57697920","6327720","332424","6561"],["259459200","3891888000","12739446720","15355327680","8505272160","2415964320","369253080","30285576","1240029","19683"],["3632428800","66940473600","266464598400","391390876800","267203059200","95578781760","19063991520","2167754400","137977830","4527090","59049"],["54486432000","1215825811200","5796577987200","10192774838400","8388655228800","3668972630400","915848065440","135231132960","11884398570","604071270","16238475","177147"],["871782912000","23264149708800","131172193152000","272209104921600","265460670921600","138904117228800","42145752116160","7751096115840","881069341680","61628260320","2559183660","57395628","531441"],["14820309504000","467898342912000","3086752891622400","7470953307417600","8514986771491200","5241313764086400","1891813675752000","420978908750400","59377131356400","5354851479120","305166412980","10547332380","200353257","1594323"],["266765571072000","9870326129664000","75485764427673600","210975997975142400","277862463066988800","198508314529324800","83829233619331200","22093618468521600","3756776243220000","419099710629600","30712542260400","1453413190320","42461073018","691936182","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","81","990","14121","229428","4170969","83707578","1835007345","43568156448","1112584528449","30380877011862","882724181789529","27174565870064460","883100568854810889"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","3"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197284,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-60a73e43da3d1ec8"]},{"id":197285,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-60a73e43da3d1ec8"]},{"id":197286,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-60a73e43da3d1ec8"]}],"region_exact":{},"incoming_edges":[{"id":"e-60a73e43da3d1ec8","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-214429b77af4dcc0","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"4883baa20ab73ee14890f24e34489ad8b9696e000413256c51859a0f4e50701b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c89ee0d57229b178","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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:         1;\n  1:         5,          3;\n  2:        30,         42,           9;\n  3:       210,        510,         243,          27;\n  4:      1680,       6240,        4932,        1188,         81;\n  5:     15120,      79920,       92700,       36180,       5265,        243;\n  6:    151200,    1084320,     1722960,      965520,     224370,      21870,       729;\n  7:   1663200,   15634080,    32543280,    24479280,    8057070,    1241730,     86751,     2187;\n  8:  19958400,  239500800,   632681280,   611694720,  266807520,   57697920,   6327720,   332424,    6561;\n  9: 259459200, 3891888000, 12739446720, 15355327680, 8505272160, 2415964320, 369253080, 30285576, 1240029, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+4+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 4 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], 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-efe23a76953cf69d","steps":["basis A013610","basis A008277"],"edges":["e-4deb1a58186d6d0a","e-60a73e43da3d1ec8"],"text":"A130534 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A130534 \u2192 basis A013610 \u2192 basis A008277","chain":null}