{"id":"n-deeca0780a643688","revision":0,"first_index":0,"fingerprint":"f6-d48d54a9c58ccb5a","name":"theta_plus_one_squared of basis_map_A013609 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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"],["3","16"],["11","120","180"],["45","744","2268","1792"],["197","4368","20160","32256","16800"],["903","25120","155520","376320","396000","152064"],["4279","143208","1112940","3611520","5742000","4447872","1345344"],["20793","813400","7609140","30997120","66066000","77261184","47087040","11714560"],["103049","4613664","50472576","247543296","662261600","1037836800","954297344","477954048","100818432"],["518859","26163504","327727296","1880126976","6056013600","11910592512","14670609408","11079843840","4701805056","859955200"],["2646723","148426200","2094789060","13763393280","51861264000","122796850176","189786455040","191942000640","122769354240","45147648000","7283820544"],["13648869","842597448","13229799660","97960558080","422796528000","1171112076288","2178881058816","2764504596480","2367178007040","1311597056000","425263060992","61345234944"],["71039373","4787262000","82769765760","681972065280","3317543064000","10526095742976","22909271024640","34986787799040","37576464330240","27865525248000","13608417951744","3943622246400","514253209600"],["372693519","27223070784","513926539776","4663733707776","25248681057600","90300313838592","225122696967168","402157705297920","519836268796416","483437574963200","315909702451200","137851506524160","36100575313920","4294166446080"],["1968801519","154943909064","3171301467516","31428891801216","187425557919600","746051733947520","2096579395694784","4290079867527168","6486189966127872","7258906566912000","5947658293857280","3472063616286720","1368813480652800","326893420707840","35739076608000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","311","4849","73781","1105927","16407163","241569237","3535900809","51513356251","747596647903","10814724222617","156017580749053","2245464447668111","32251158082220899"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","-6","8","0","0"],["2","-3","-16","-6","0"],["0","0","-30","-9","4"],["0","0","-10","-3","10"],["0","0","0","0","8"],["0","0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["12","18","-48","-112","-48","16"],["6","37","-108","-322","-120","72"],["0","14","-102","-287","-84","128"],["0","0","-30","-93","6","112"],["0","0","0","-8","24","48"],["0","0","0","0","6","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","0","-19","-48","-38","0","8"],["0","0","-28","-96","-76","24","32"],["0","0","-14","-48","-22","60","48"],["0","0","0","0","16","48","32"],["0","0","0","0","4","12","8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185659,"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-74e6da031cd6ca35"]},{"id":185660,"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-74e6da031cd6ca35"]},{"id":185661,"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-74e6da031cd6ca35"]},{"id":185662,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-74e6da031cd6ca35"]}],"region_exact":{},"incoming_edges":[{"id":"e-74e6da031cd6ca35","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-deeca0780a643688","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"2261b90f1af4aa84c64d587d08bd765dbd658a7ee975306b6cce9c5677bf2a21","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8217f7f3de935aeb","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:      3,       16;\n  2:     11,      120,       180;\n  3:     45,      744,      2268,       1792;\n  4:    197,     4368,     20160,      32256,      16800;\n  5:    903,    25120,    155520,     376320,     396000,      152064;\n  6:   4279,   143208,   1112940,    3611520,    5742000,     4447872,     1345344;\n  7:  20793,   813400,   7609140,   30997120,   66066000,    77261184,    47087040,    11714560;\n  8: 103049,  4613664,  50472576,  247543296,  662261600,  1037836800,   954297344,   477954048,  100818432;\n  9: 518859, 26163504, 327727296, 1880126976, 6056013600, 11910592512, 14670609408, 11079843840, 4701805056, 859955200;\n","recurrence_sentence":"The row polynomials satisfy (2*n+4-(3*n+6)*x-(10*n^3+30*n^2+16*n-8)*x^2-(3*n^3+9*n^2+6*n)*x^3+(2*n^5+8*n^4+10*n^3+4*n^2)*x^4)*P(n,x) = (6*n+12+(14*n^2+37*n+18)*x-(30*n^3+102*n^2+108*n+48)*x^2-(8*n^4+93*n^3+287*n^2+322*n+112)*x^3+(6*n^5+24*n^4+6*n^3-84*n^2-120*n-48)*x^4+(8*n^5+48*n^4+112*n^3+128*n^2+72*n+16)*x^5)*P(n-1,x) + (1-(14*n^2+28*n+19)*x^2-(48*n^2+96*n+48)*x^3+(4*n^4+16*n^3-22*n^2-76*n-38)*x^4+(12*n^4+48*n^3+60*n^2+24*n)*x^5+(8*n^4+32*n^3+48*n^2+32*n+8)*x^6)*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[Together[((6*n + 12 + (14*n^2 + 37*n + 18)*x - (30*n^3 + 102*n^2 + 108*n + 48)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (6*n^5 + 24*n^4 + 6*n^3 - 84*n^2 - 120*n - 48)*x^4 + (8*n^5 + 48*n^4 + 112*n^3 + 128*n^2 + 72*n + 16)*x^5)*P[n - 1, x] + (1 - (14*n^2 + 28*n + 19)*x^2 - (48*n^2 + 96*n + 48)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (12*n^4 + 48*n^3 + 60*n^2 + 24*n)*x^5 + (8*n^4 + 32*n^3 + 48*n^2 + 32*n + 8)*x^6)*D[P[n - 1, x], x])/(2*n + 4 - (3*n + 6)*x - (10*n^3 + 30*n^2 + 16*n - 8)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^4)]];\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] = ((6*n + 12 + (14*n^2 + 37*n + 18)*x - (30*n^3 + 102*n^2 + 108*n + 48)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (6*n^5 + 24*n^4 + 6*n^3 - 84*n^2 - 120*n - 48)*x^4 + (8*n^5 + 48*n^4 + 112*n^3 + 128*n^2 + 72*n + 16)*x^5)*v[i - 1] + (1 - (14*n^2 + 28*n + 19)*x^2 - (48*n^2 + 96*n + 48)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (12*n^4 + 48*n^3 + 60*n^2 + 24*n)*x^5 + (8*n^4 + 32*n^3 + 48*n^2 + 32*n + 8)*x^6)*deriv(v[i - 1], x))/(2*n + 4 - (3*n + 6)*x - (10*n^3 + 30*n^2 + 16*n - 8)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*n^2)*x^4)); 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-66e2c3448245b0b7","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-4e2df72103232439","e-74e6da031cd6ca35"],"text":"A033282 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A033282 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}