{"id":"n-3c97857fd579ea19","revision":1,"first_index":0,"fingerprint":"f6-1116965b4148f56c","name":"one_plus_derivative 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"],["3","2"],["6","15","5"],["10","51","63","14"],["15","126","308","252","42"],["21","260","1020","1620","990","132"],["28","477","2700","6765","7920","3861","429"],["36","805","6160","21945","40040","37037","15015","1430"],["45","1276","12628","60060","154154","220220","168168","58344","4862"],["55","1926","23868","145236","493038","982800","1150968","747864","226746","16796"],["66","2795","42315","319410","1375920","3601416","5847660","5794620","3275220","881790","58786"],["78","3927","71225","651420","3455760","11404008","24174612","33042900","28352940","14176470","3432198","208012"],["91","5370","114840","1249160","7977420","32232816","85698360","152210520","179390970","135653540","60798936","13372200","742900"],["105","7176","178568","2275416","17182308","83140200","269374248","597500904","911493726","943364136","637408200","258826360","52151580","2674440"],["120","9401","269178","3968055","34918884","198843645","768215448","2068389609","3926434512","5242682445","4833678850","2950910235","1095183180","203591745","9694845"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","138","743","4043","22180","122468","679757","3789297","21199998","118973550","669447123","3775577367","21336790152"],"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","6","-2"],["2","3","-1"],["3","0","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["4","10","6","-4"],["8","17","3","-2"],["5","10","0","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","4","0","-2"],["0","4","4","0","0"],["0","2","2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164915,"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-9bda39d02f544942"]},{"id":164916,"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-9bda39d02f544942"]},{"id":164917,"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-9bda39d02f544942"]},{"id":164918,"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-9bda39d02f544942"]}],"region_exact":{},"incoming_edges":[{"id":"e-9bda39d02f544942","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-3c97857fd579ea19","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":"9d3fa4a73002015839ae2a3dc4cf4534b0160aae41a15b584dc7d1d178502e0a","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-66e2c3448245b0b7","role":"sequence","name":"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":[{"id":"l-26fb710194da6545","kind":"oeis_match","node_a_id":"n-3c97857fd579ea19","node_b_id":null,"fingerprint":null,"oeis_id":"A035029","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A035029","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A035029","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3c97857fd579ea19","oeis_id":"A035029","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":16081}],"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:  6,   15,     5;\n  3: 10,   51,    63,     14;\n  4: 15,  126,   308,    252,     42;\n  5: 21,  260,  1020,   1620,    990,    132;\n  6: 28,  477,  2700,   6765,   7920,   3861,     429;\n  7: 36,  805,  6160,  21945,  40040,  37037,   15015,   1430;\n  8: 45, 1276, 12628,  60060, 154154, 220220,  168168,  58344,   4862;\n  9: 55, 1926, 23868, 145236, 493038, 982800, 1150968, 747864, 226746, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n+(3*n+6)*x-(n+2)*x^2)*P(n,x) = (n^3+5*n^2+8*n+4+(2*n^3+10*n^2+17*n+10)*x+(3*n+6)*x^2-(2*n+4)*x^3)*P(n-1,x) + ((2*n^2+4*n+2)*x+(2*n^2+4*n+4)*x^2-2*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 + 5*n^2 + 8*n + 4 + (2*n^3 + 10*n^2 + 17*n + 10)*x + (3*n + 6)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + ((2*n^2 + 4*n + 2)*x + (2*n^2 + 4*n + 4)*x^2 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (3*n + 6)*x - (n + 2)*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 + 5*n^2 + 8*n + 4 + (2*n^3 + 10*n^2 + 17*n + 10)*x + (3*n + 6)*x^2 - (2*n + 4)*x^3)*v[i - 1] + ((2*n^2 + 4*n + 2)*x + (2*n^2 + 4*n + 4)*x^2 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (3*n + 6)*x - (n + 2)*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":["D+1"],"edges":["e-9bda39d02f544942"],"text":"A033282 \u2192 D+1"},"headline":"A033282 \u2192 D+1","chain":null}
