{"id":"n-5ef5d1e2cb1d41ee","revision":0,"first_index":0,"fingerprint":"f6-2fb8ba1337b01a08","name":"one_plus_derivative of theta_plus_one 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"],["5","4"],["11","40","15"],["19","144","231","56"],["29","364","1176","1176","210"],["41","760","3960","7800","5610","792"],["55","1404","10575","33000","46035","25740","3003"],["71","2380","24255","107800","235235","252252","115115","11440"],["89","3784","49896","296296","910910","1513512","1313312","505648","43758"],["109","5724","94536","718536","2923830","6787872","9059232","6587568","2191878","167960"],["131","8320","167895","1583400","8179080","24950016","46224360","51395760","32122350","9405760","646646"],["155","11704","282975","3234000","20577480","79168320","191606184","294188400","279798750","153205360","40042310","2496144"],["181","16020","456720","6208400","47561580","224094816","680470560","1358382960","1776322350","1474210760","717819696","169381200","9657700"],["209","21424","710736","11318736","102539580","578655792","2141691552","5340926448","9044822358","10283518960","7563910640","3314522640","712738260","37442160"],["239","28084","1072071","19752096","208543335","1385115732","6113714607","18510334128","39018942963","57260503300","57520778315","37966350240","15119612235","2986012260","145422675"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","66","450","2955","18963","119812","748548","4637205","28537245","174683718","1064611782","6464582943","39132819495","236256182280"],"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":[["0","0","0"],["-2","6","-2"],["1","3","-1"],["3","0","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","8","-4"],["2","18","10","-6"],["7","25","3","-2"],["5","12","0","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","0","-2"],["0","4","6","0","-2"],["0","6","6","0","0"],["0","2","2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178638,"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-9b07c07abd3fcf72"]},{"id":178639,"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-9b07c07abd3fcf72"]},{"id":178640,"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-9b07c07abd3fcf72"]}],"region_exact":{},"incoming_edges":[{"id":"e-9b07c07abd3fcf72","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-5ef5d1e2cb1d41ee","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":"674163adba7aecb6fe957860e54e1b4c3dcc0a3cfe8dc21b6d873c02954b0ef8","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-7b4f74988046113d","role":"sequence","name":"theta_plus_one 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:   5,    4;\n  2:  11,   40,    15;\n  3:  19,  144,   231,     56;\n  4:  29,  364,  1176,   1176,     210;\n  5:  41,  760,  3960,   7800,    5610,     792;\n  6:  55, 1404, 10575,  33000,   46035,   25740,    3003;\n  7:  71, 2380, 24255, 107800,  235235,  252252,  115115,   11440;\n  8:  89, 3784, 49896, 296296,  910910, 1513512, 1313312,  505648,   43758;\n  9: 109, 5724, 94536, 718536, 2923830, 6787872, 9059232, 6587568, 2191878, 167960;\n","recurrence_sentence":"The row polynomials satisfy (n^4+3*n^3+n^2-2*n+(3*n^2+6*n)*x-(n^2+2*n)*x^2)*P(n,x) = (n^4+5*n^3+7*n^2+2*n+(2*n^4+12*n^3+25*n^2+18*n)*x+(3*n^2+10*n+8)*x^2-(2*n^2+6*n+4)*x^3)*P(n-1,x) + ((2*n^3+6*n^2+4*n)*x+(2*n^3+6*n^2+6*n+2)*x^2-(2*n+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^4 + 5*n^3 + 7*n^2 + 2*n + (2*n^4 + 12*n^3 + 25*n^2 + 18*n)*x + (3*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 4)*x^3)*P[n - 1, x] + ((2*n^3 + 6*n^2 + 4*n)*x + (2*n^3 + 6*n^2 + 6*n + 2)*x^2 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 + 3*n^3 + n^2 - 2*n + (3*n^2 + 6*n)*x - (n^2 + 2*n)*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^4 + 5*n^3 + 7*n^2 + 2*n + (2*n^4 + 12*n^3 + 25*n^2 + 18*n)*x + (3*n^2 + 10*n + 8)*x^2 - (2*n^2 + 6*n + 4)*x^3)*v[i - 1] + ((2*n^3 + 6*n^2 + 4*n)*x + (2*n^3 + 6*n^2 + 6*n + 2)*x^2 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 + 3*n^3 + n^2 - 2*n + (3*n^2 + 6*n)*x - (n^2 + 2*n)*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-66e2c3448245b0b7","steps":["\u03b8+1","D+1"],"edges":["e-10c4942a23949b48","e-9b07c07abd3fcf72"],"text":"A033282 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A033282 \u2192 \u03b8+1 \u2192 D+1","chain":null}