{"id":"n-6273e0e84ee46315","revision":1,"first_index":0,"fingerprint":"f6-f72e0f3565a42e07","name":"derivative of Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 0 <= 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":true,"rows":[[],["3"],["8","10"],["15","36","21"],["24","84","96","36"],["35","160","270","200","55"],["48","270","600","660","360","78"],["63","420","1155","1680","1365","588","105"],["80","616","2016","3640","3920","2520","896","136"],["99","864","3276","7056","9450","8064","4284","1296","171"],["120","1170","5040","12600","20160","21420","15120","6840","1800","210"],["143","1540","7425","21120","39270","49896","43890","26400","10395","2420","253"],["168","1980","10560","33660","71280","105336","110880","83160","43560","15180","3168","300"],["195","2496","14586","51480","122265","205920","252252","226512","148005","68640","21450","4056","351"],["224","3094","19656","76076","200200","378378","528528","552552","432432","250250","104104","29484","5096","406"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","18","72","240","720","2016","5376","13824","34560","84480","202752","479232","1118208","2580480"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["-1","1"],["0","-3"],["1","2"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["2","3","1"],["1","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163440,"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-61bfd1b71b029f2e"]},{"id":163441,"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-61bfd1b71b029f2e"]},{"id":163442,"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-61bfd1b71b029f2e"]}],"region_exact":{},"incoming_edges":[{"id":"e-61bfd1b71b029f2e","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-6273e0e84ee46315","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":"690fa82d0c9a54b43ccc16a63d0f7264d1d8f5c394d710c18719144588492401","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-0486a9621eaf19a3","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+k+1)*binomial(n,k), 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-d2278cf2c774ca82","kind":"fingerprint_related","node_a_id":"n-b73c4ec16797bcf3","node_b_id":"n-6273e0e84ee46315","fingerprint":"f6-f72e0f3565a42e07","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f72e0f3565a42e07"},"other_id":"n-b73c4ec16797bcf3","other":{"id":"n-b73c4ec16797bcf3","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-f72e0f3565a42e07","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["5","4"],["14","24","10"],["30","80","70","20"]],"card_rows":[["5","4"],["14","24","10"],["30","80","70","20"],["55","200","270","160","35"]],"row_sums_preview":["9","48","200","720","2352","7168","20736","57600"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-ac667fe3dc6003e4"],"text":"A178820 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A178820 \u2192 P\u2099 + P\u2099\u208a\u2081","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:  0;\n  1:  3;\n  2:  8,  10;\n  3: 15,  36,   21;\n  4: 24,  84,   96,   36;\n  5: 35, 160,  270,  200,   55;\n  6: 48, 270,  600,  660,  360,   78;\n  7: 63, 420, 1155, 1680, 1365,  588,  105;\n  8: 80, 616, 2016, 3640, 3920, 2520,  896,  136;\n  9: 99, 864, 3276, 7056, 9450, 8064, 4284, 1296, 171;\n","recurrence_sentence":"The row polynomials satisfy (n^2-1+(2*n^2-3*n+1)*x)*P(n,x) = (n^2+2*n+(3*n^2+3*n)*x+(2*n^2+n)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (3*n^2 + 3*n)*x + (2*n^2 + n)*x^2)*P[n - 1, x])/(n^2 - 1 + (2*n^2 - 3*n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (3*n^2 + 3*n)*x + (2*n^2 + n)*x^2)*v[i - 1])/(n^2 - 1 + (2*n^2 - 3*n + 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-c8c19eba9d68881f","steps":["\u03b8+1+n","D"],"edges":["e-dd616c2789f09167","e-61bfd1b71b029f2e"],"text":"A007318 \u2192 \u03b8+1+n \u2192 D"},"headline":"A007318 \u2192 \u03b8+1+n \u2192 D","chain":null}
