{"id":"n-7cb278ca98b33a0d","revision":0,"first_index":0,"fingerprint":"f6-ff3c128a3f309512","name":"derivative of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["12","2"],["103","138","3"],["834","4340","948","4"],["6685","107820","83490","5316","5"],["53496","2438030","4956900","1124780","27060","6"],["427987","52910502","245817585","139202700","12439585","130686","7"],["3423918","1127384048","11102555448","13309013640","2934233910","121873752","611100","8"],["27391369","23825717400","476232663288","1101940402824","497957927010","51420015936","1102855908","2795400","9"],["219130980","501709633850","19835555205720","83690686986856","69842769049440","14607005780844","794002825728","9451812264","12581820","10"],["1753047871","10548173645730","812236319498175","6021737606846760","8692559644684070","3283712934601500","361284085004262","11200975767720","77955931179","55922610","11"],["14024383002","221620335528332","32932476073047660","418487488428311900","1000083510077956620","634924180200796296","124881973739978112","7887868810657464","147838638116970","625393519180","246063972","12"],["112195064053","4654980704139108","1327604388523973838","28431165854567386140","109085750980868582395","110847120632763962136","36053380039353798940","4058326573643243448","156754900028617011","1855910724673380","4915924198262","1073739468","13"],["897560512464","97762897221661190","53341579556870048028","1902900095272537684404","11467322899073242424820","18015239573583664435938","9190470983167210308264","1695996874209983695640","116941250333591145144","2897103001971423850","22420715044298076","38056253470836","4652878308","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","14","244","6126","203316","8600272","450929052","28599095824","2152509769144","189281413007512","19193359504949888","2219567595968359520","289966619290371378192","42445230580769803986976"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-5"],["3"]]},"terms":[{"coeff":{"coeffs":[["16","-32","80"],["-24","60","-198"],["0","-18","135"],["0","0","-27"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","26","-106","30"],["0","-39","219","-63"],["0","0","-90","27"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","6","-30"],["0","0","-9","63"],["0","0","0","-27"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-42","50"],["27","-60"],["0","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-65","15"],["0","39","-9"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-15"],["0","0","9"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189613,"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-907caf4394cc8d57"]},{"id":189614,"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-907caf4394cc8d57"]},{"id":189615,"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-907caf4394cc8d57"]}],"region_exact":{},"incoming_edges":[{"id":"e-907caf4394cc8d57","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7cb278ca98b33a0d","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"4939d79a0ea0cd14abe0493ca9912b0699f9a60122fb28a8656e3cc670c408a2","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-25ac1cc25c118491","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=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:        0;\n  1:        1;\n  2:       12,           2;\n  3:      103,         138,            3;\n  4:      834,        4340,          948,             4;\n  5:     6685,      107820,        83490,          5316,            5;\n  6:    53496,     2438030,      4956900,       1124780,        27060,           6;\n  7:   427987,    52910502,    245817585,     139202700,     12439585,      130686,          7;\n  8:  3423918,  1127384048,  11102555448,   13309013640,   2934233910,   121873752,     611100,       8;\n  9: 27391369, 23825717400, 476232663288, 1101940402824, 497957927010, 51420015936, 1102855908, 2795400, 9;\n","recurrence_sentence":"The row polynomials satisfy (3*n-5)*P(n,x) = -(24*n-16+(18*n^2-60*n+32)*x+(27*n^3-135*n^2+198*n-80)*x^2)*P(n-2,x) - ((39*n-26)*x+(90*n^2-219*n+106)*x^2-(27*n^2-63*n+30)*x^3)*d/dx P(n-2,x) - ((9*n-6)*x^2+(27*n^2-63*n+30)*x^3)*d^2/dx^2 P(n-2,x) + (27*n-42+(18*n^2-60*n+50)*x)*P(n-1,x) + ((39*n-65)*x-(9*n-15)*x^2)*d/dx P(n-1,x) + (9*n-15)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 16 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*P[n - 2, x] - ((39*n - 26)*x + (90*n^2 - 219*n + 106)*x^2 - (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], x] - ((9*n - 6)*x^2 + (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], {x, 2}] + (27*n - 42 + (18*n^2 - 60*n + 50)*x)*P[n - 1, x] + ((39*n - 65)*x - (9*n - 15)*x^2)*D[P[n - 1, x], x] + (9*n - 15)*x^2*D[P[n - 1, x], {x, 2}])/(3*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n - 16 + (18*n^2 - 60*n + 32)*x + (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*v[i - 2] - ((39*n - 26)*x + (90*n^2 - 219*n + 106)*x^2 - (27*n^2 - 63*n + 30)*x^3)*deriv(v[i - 2], x) - ((9*n - 6)*x^2 + (27*n^2 - 63*n + 30)*x^3)*deriv(deriv(v[i - 2], x), x) + (27*n - 42 + (18*n^2 - 60*n + 50)*x)*v[i - 1] + ((39*n - 65)*x - (9*n - 15)*x^2)*deriv(v[i - 1], x) + (9*n - 15)*x^2*deriv(deriv(v[i - 1], x), x))/(3*n - 5)); 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-25ac1cc25c118491","steps":["D"],"edges":["e-907caf4394cc8d57"],"text":"A166962 \u2192 D"},"headline":"A166962 \u2192 D","chain":null}