{"id":"n-3d889b8995061d11","revision":0,"first_index":0,"fingerprint":"f6-d6e98125c0869a94","name":"basis_map_A008277 of reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["6","2"],["24","24","4"],["120","264","116","12"],["720","3120","2680","680","48"],["5040","41040","61160","28360","4696","240"],["40320","604800","1469440","1104320","320432","37296","1440"],["362880","9959040","38030720","43415680","19586224","3884048","334512","10080"],["3628800","182165760","1068471040","1780378880","1169946848","350892192","50474080","3341376","80640"],["39916800","3675974400","32625152000","77270636800","71010911776","30179484384","6438844928","701528544","36765504","725760"],["479001600","81270604800","1080764608000","3571618265600","4465273780352","2581237149568","765391678656","121913789888","10397711808","441682560","7257600"],["6227020800","1955763532800","38720737932800","176182850905600","293808406401664","224642137754496","88859551259072","19391549555136","2391220581376","163843276608","5751192960","79833600"],["87178291200","50924576102400","1494857203840000","9273791400473600","20330315407770368","20149219070652672","10337627061334144","2966686471512192","495732858612992","48683526722688","2736779175936","80670021120","958003200"],["1307674368000","1427021448652800","61951164846771200","520217879275494400","1482690420526771456","1876231625953228544","1223339164622560128","448857917777636224","97559134304604544","12870752689363712","1029718827905664","48323992624128","1212528821760","12454041600"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","2","8","52","512","7248","140536","3578048","115583184","4609379616","221979940896","12678795530432","846128119246912","65150743162512512","5726225656922844160"],"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":[["1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196726,"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-f27b44e863174b8a"]},{"id":196727,"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-f27b44e863174b8a"]},{"id":196728,"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-f27b44e863174b8a"]}],"region_exact":{},"incoming_edges":[{"id":"e-f27b44e863174b8a","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3d889b8995061d11","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":"505fc3150ab1c66509fe34a106e1edf8b92e30841a6eebb96d6d67cb3c6643dd","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-3df3926aefc1fd76","role":"sequence","name":"reversal of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:       2;\n  2:       6,         2;\n  3:      24,        24,          4;\n  4:     120,       264,        116,         12;\n  5:     720,      3120,       2680,        680,         48;\n  6:    5040,     41040,      61160,      28360,       4696,       240;\n  7:   40320,    604800,    1469440,    1104320,     320432,     37296,     1440;\n  8:  362880,   9959040,   38030720,   43415680,   19586224,   3884048,   334512,   10080;\n  9: 3628800, 182165760, 1068471040, 1780378880, 1169946848, 350892192, 50474080, 3341376, 80640;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+(n-1)*x)*P(n-1,x) + (n-1)*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 + 1 + (n - 1)*x)*P[n - 1, x] + (n - 1)*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 + 1 + (n - 1)*x)*v[i - 1] + (n - 1)*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-60c7e36cbe887b7e","steps":["reverse","basis A008277"],"edges":["e-a28ae8b138b0cf1e","e-f27b44e863174b8a"],"text":"A137320 \u2192 reverse \u2192 basis A008277"},"headline":"A137320 \u2192 reverse \u2192 basis A008277","chain":null}