{"id":"n-c5d3384acf72de77","revision":1,"first_index":0,"fingerprint":"f6-770cb16f18256565","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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","2"],["7","14","4"],["13","60","48","8"],["21","180","320","144","16"],["31","430","1400","1360","400","32"],["43","882","4620","8120","5040","1056","64"],["57","1624","12544","35280","39200","17024","2688","128"],["73","2760","29568","122304","216384","166656","53760","6656","256"],["91","4410","62640","358848","931392","1137024","645120","161280","16128","512"],["111","6710","122100","926640","3326400","5943168","5322240","2323200","464640","38400","1024"],["133","9812","222640","2163480","10280160","25369344","33175296","22767360","7898880","1295360","90112","2048"],["157","13884","384384","4656080","28314000","92416896","166712832","166712832","90604800","25625600","3514368","208896","4096"],["183","19110","634088","9369360","70998928","296428704","706717440","973699584","769536768","339795456","79951872","9318400","479232","8192"],["211","25690","1006460","17820712","164692528","856567712","2610928320","4750711680","5166889728","3311340032","1212603392","241346560","24227840","1089536","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","25","129","681","3653","19825","108545","598417","3317445","18474633","103274625","579168825","3256957317","18359266785"],"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":[["4","-10","4"],["-2","3","2"],["2","4","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["4","-10","4","0"],["6","1","-16","12"],["6","0","24","0"],["2","12","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-6","8","-8"],["0","2","4","-16","0"],["0","4","-8","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165269,"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-9941cb9a817a797c"]},{"id":165270,"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-9941cb9a817a797c"]},{"id":165271,"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-9941cb9a817a797c"]},{"id":165272,"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-9941cb9a817a797c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9941cb9a817a797c","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-c5d3384acf72de77","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":"1ba04c95d2b1480e8bf735465563f251232db9c9d5ea3770b945fd5058b195c2","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-8ce61d302cec295b","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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-9a661fcfb14c596c","kind":"oeis_match","node_a_id":"n-c5d3384acf72de77","node_b_id":null,"fingerprint":null,"oeis_id":"A002002","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002002","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A002002","relation":"match_row_sums"}],"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,    2;\n  2:  7,   14,     4;\n  3: 13,   60,    48,      8;\n  4: 21,  180,   320,    144,     16;\n  5: 31,  430,  1400,   1360,    400,      32;\n  6: 43,  882,  4620,   8120,   5040,    1056,     64;\n  7: 57, 1624, 12544,  35280,  39200,   17024,   2688,    128;\n  8: 73, 2760, 29568, 122304, 216384,  166656,  53760,   6656,   256;\n  9: 91, 4410, 62640, 358848, 931392, 1137024, 645120, 161280, 16128, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+2*n^2-2*n+4+(4*n^2+3*n-10)*x+(2*n+4)*x^2)*P(n,x) = (2*n^3+6*n^2+6*n+4+(12*n^3+n-10)*x+(24*n^2-16*n+4)*x^2+12*n*x^3)*P(n-1,x) + ((4*n^2+2*n+2)*x-(8*n^2-4*n+6)*x^2-(16*n-8)*x^3-8*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[((2*n^3 + 6*n^2 + 6*n + 4 + (12*n^3 + n - 10)*x + (24*n^2 - 16*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] + ((4*n^2 + 2*n + 2)*x - (8*n^2 - 4*n + 6)*x^2 - (16*n - 8)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 2*n^2 - 2*n + 4 + (4*n^2 + 3*n - 10)*x + (2*n + 4)*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] = ((2*n^3 + 6*n^2 + 6*n + 4 + (12*n^3 + n - 10)*x + (24*n^2 - 16*n + 4)*x^2 + 12*n*x^3)*v[i - 1] + ((4*n^2 + 2*n + 2)*x - (8*n^2 - 4*n + 6)*x^2 - (16*n - 8)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 2*n^2 - 2*n + 4 + (4*n^2 + 3*n - 10)*x + (2*n + 4)*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-8ce61d302cec295b","steps":["D+1"],"edges":["e-9941cb9a817a797c"],"text":"A114687 \u2192 D+1"},"headline":"A114687 \u2192 D+1","chain":null}
