{"id":"n-854b531da9a10af4","revision":0,"first_index":0,"fingerprint":"f6-695598672c6e8bd2","name":"one_plus_derivative of Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.","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"],["4","2"],["24","42","12"],["168","624","600","120"],["1320","8400","16200","10920","1680"],["11520","111600","352800","463680","241920","30240"],["110880","1517040","7056000","14817600","14817600","6320160","665280"],["1169280","21450240","137168640","414892800","651974400","526901760","190270080","17297280"],["13426560","317882880","2667168000","10897286400","24309331200","30512401920","20704844160","6486480000","518918400"],["166924800","4953312000","52690176000","278299929600","828193766400","1460236377600","1525620096000","892539648000","247005158400","17643225600"],["2235340800","81230688000","1066976064000","7055214566400","26850913689600","62572218508800","91101314304000","81511702272000","41920304025600","10391859878400","670442572800"],["32093107200","1401558681600","22261599360000","179805225600000","848680664832000","2508642507571200","4792888093593600","5938709736960000","4646167029504000","2132007381504000","478695996979200","28158588057600"],["491934643200","25418698905600","479960082201600","4643769626496000","26553635716608000","96640274213683200","232699607646105600","376206781760409600","405544007860992000","281867466456576000","116773664674867200","23962958437017600","1295295050649600"],["8020402790400","483926694451200","10709853073920000","122171657287680000","830767269556224000","3639269273678438400","10727070437718835200","21744061698078720000","30441686377310208000","29032349045027328000","18152264839803494400","6854701408037683200","1295295050649600000","64764752532480000"],["138613483008000","9657175207680000","247516604375040000","3285306929608704000","26169168991021056000","135537984584690688000","478369357357731840000","1181427352262277120000","2064961059260875776000","2547800029300990464000","2178271780776419328000","1237654420895692800000","429390309290342400000","75191877690209280000","3497296636753920000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","78","1512","38520","1211760","45304560","1961124480","96427739520","5307348614400","323224411910400","21577450475750400","1566693332159155200","122915603609851852800","10361813256369679104000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","1","-1"],["-1","0","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["2","1","-1","0"],["3","3","-1","-2"],["2","2","0","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","0","-4","-2"],["0","0","0","0","0"],["0","2","2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190860,"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-b2374e2588a9a86b"]},{"id":190861,"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-b2374e2588a9a86b"]},{"id":190862,"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-b2374e2588a9a86b"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2374e2588a9a86b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-854b531da9a10af4","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":"2998b8341bafd5f8ef3516c9117de42c33f9711c1eb4208c88b551b8c5689a17","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-694017a14c5c1f2e","role":"sequence","name":"Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with 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":[{"id":"l-53f36b69d3dec681","kind":"fingerprint_related","node_a_id":"n-cc0179e01d57c43d","node_b_id":"n-854b531da9a10af4","fingerprint":"f6-695598672c6e8bd2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-695598672c6e8bd2"},"other_id":"n-cc0179e01d57c43d","other":{"id":"n-cc0179e01d57c43d","name":"one_plus_derivative of Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.","first_index":1,"fingerprint":"f6-695598672c6e8bd2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","7","2"]],"card_rows":[["1"],["2","1"],["4","7","2"],["7","26","25","5"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-be392fc9b98fb771","steps":["D+1"],"edges":["e-551a859050a73480"],"text":"A088617 \u2192 D+1"},"headline":"A088617 \u2192 D+1","chain":null}}],"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:         4,          2;\n  2:        24,         42,          12;\n  3:       168,        624,         600,          120;\n  4:      1320,       8400,       16200,        10920,         1680;\n  5:     11520,     111600,      352800,       463680,       241920,         30240;\n  6:    110880,    1517040,     7056000,     14817600,     14817600,       6320160,        665280;\n  7:   1169280,   21450240,   137168640,    414892800,    651974400,     526901760,     190270080,     17297280;\n  8:  13426560,  317882880,  2667168000,  10897286400,  24309331200,   30512401920,   20704844160,   6486480000,    518918400;\n  9: 166924800, 4953312000, 52690176000, 278299929600, 828193766400, 1460236377600, 1525620096000, 892539648000, 247005158400, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n+2+x-x^2)*P(n,x) = (n^3+2*n^2+3*n+2+(2*n^3+2*n^2+3*n+1)*x-(n+1)*x^2-2*n*x^3)*P(n-1,x) + ((2*n^2+2)*x+2*n^2*x^2-4*x^3-2*x^4)*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[((n^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*P[n - 1, x] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^2 - n + 2 + x - x^2)]];\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^3 + 2*n^2 + 3*n + 2 + (2*n^3 + 2*n^2 + 3*n + 1)*x - (n + 1)*x^2 - 2*n*x^3)*v[i - 1] + ((2*n^2 + 2)*x + 2*n^2*x^2 - 4*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^2 - n + 2 + x - x^2)); 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-694017a14c5c1f2e","steps":["D+1"],"edges":["e-b2374e2588a9a86b"],"text":"A253284 \u2192 D+1"},"headline":"A253284 \u2192 D+1","chain":null}
