{"id":"n-7088e01c47610317","revision":0,"first_index":0,"fingerprint":"f6-74a850e0b5b3645e","name":"one_plus_derivative of veronese_section_2_1 of Triangle of coefficients in expansion of (1+2*x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["4"],["14","8"],["40","32"],["90","144","32"],["172","544","192"],["294","1624","1056","128"],["464","4032","4864","1024"],["690","8736","17856","6656","512"],["980","17088","54144","35840","5120"],["1342","30888","141504","154880","38400","2048"],["1784","52448","329472","551936","235520","24576"],["2314","84656","700128","1683968","1164800","208896","8192"],["2940","131040","1381952","4539392","4752384","1433600","114688"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","4","22","72","266","908","3102","10384","34450","113172","369062","1195736","3852954","12355996"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-15","30","-8"],["25","-38","8"],["-12","8","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-72","176","-64"],["7","86","-144","32"],["-6","-48","32","0"],["2","8","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10","-68","128","-64"],["0","-12","64","-64","0"],["0","4","-16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173607,"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-0e8cacbe03cce126"]},{"id":173608,"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-0e8cacbe03cce126"]},{"id":173609,"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-0e8cacbe03cce126"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e8cacbe03cce126","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-7088e01c47610317","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":"d3afe37308ac6abf0c109e1cd43b05832ef319cca037e7ea3c793133aae2d1b5","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-1864d83319c9138c","role":"sequence","name":"veronese_section_2_1 of Triangle of coefficients in expansion of (1+2*x)^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-36922974c7ae9c89","kind":"fingerprint_related","node_a_id":"n-7088e01c47610317","node_b_id":"n-4d16118b0971adf3","fingerprint":"f6-74a850e0b5b3645e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74a850e0b5b3645e"},"other_id":"n-4d16118b0971adf3","other":{"id":"n-4d16118b0971adf3","name":"one_plus_derivative of veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-74a850e0b5b3645e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["12"]],"card_rows":[["6"],["12"],["42","24"],["120","96"]],"row_sums_preview":["0","6","12","66","216","798","2724","9306"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-1dcfbe1f89fd593f","e-8e17c2478c848313"],"text":"A120909 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A120909 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}},{"id":"l-6c746a3b468d309c","kind":"fingerprint_related","node_a_id":"n-92abebe007fb10d9","node_b_id":"n-7088e01c47610317","fingerprint":"f6-74a850e0b5b3645e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74a850e0b5b3645e"},"other_id":"n-92abebe007fb10d9","other":{"id":"n-92abebe007fb10d9","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","first_index":0,"fingerprint":"f6-74a850e0b5b3645e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["7","4"]],"card_rows":[["1"],["2"],["7","4"],["20","16"]],"row_sums_preview":["1","2","11","36","133","454","1551","5192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf38558d99f4fe37","steps":["D+1"],"edges":["e-7e72e7621d3378c8"],"text":"A229032 \u2192 D+1"},"headline":"A229032 \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:   0;\n  1:   2;\n  2:   4;\n  3:  14,    8;\n  4:  40,   32;\n  5:  90,  144,    32;\n  6: 172,  544,   192;\n  7: 294, 1624,  1056,  128;\n  8: 464, 4032,  4864, 1024;\n  9: 690, 8736, 17856, 6656, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-12*n^2+25*n-15+(8*n^2-38*n+30)*x+(8*n-8)*x^2)*P(n,x) = (2*n^3-6*n^2+7*n+(8*n^3-48*n^2+86*n-72)*x+(32*n^2-144*n+176)*x^2+(32*n-64)*x^3)*P(n-1,x) + ((4*n^2-12*n+10)*x-(16*n^2-64*n+68)*x^2-(64*n-128)*x^3-64*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 6*n^2 + 7*n + (8*n^3 - 48*n^2 + 86*n - 72)*x + (32*n^2 - 144*n + 176)*x^2 + (32*n - 64)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 10)*x - (16*n^2 - 64*n + 68)*x^2 - (64*n - 128)*x^3 - 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 12*n^2 + 25*n - 15 + (8*n^2 - 38*n + 30)*x + (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^3 - 6*n^2 + 7*n + (8*n^3 - 48*n^2 + 86*n - 72)*x + (32*n^2 - 144*n + 176)*x^2 + (32*n - 64)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 10)*x - (16*n^2 - 64*n + 68)*x^2 - (64*n - 128)*x^3 - 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 12*n^2 + 25*n - 15 + (8*n^2 - 38*n + 30)*x + (8*n - 8)*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-71965e79e47a4526","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-c6792048648d3d6d","e-0e8cacbe03cce126"],"text":"A013609 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A013609 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}
