{"id":"n-16a684c04a1ed263","revision":0,"first_index":1,"fingerprint":"f6-58fd21a6ca87df0d","name":"derivative of shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["16","4"],["102","60","6"],["608","584","144","8"],["3530","4760","1980","280","10"],["20256","35340","21120","5160","480","12"],["115598","248052","195090","69720","11410","756","14"],["658048","1678096","1644384","784112","190400","22512","1120","16"],["3742290","11064240","13015800","7817040","2561580","454608","40824","1584","18"],["21278640","71591700","98438400","71593200","29820000","7209720","981120","69360","2160","20"],["121023606","456687660","719358750","615872400","312981900","96936840","18115020","1956240","111870","2860","22"],["688686048","2881075032","5118228720","5049574200","3039696000","1156169520","278641440","41603760","3659040","172920","3696","24"],["3921503898","18015323976","35647444404","39862683432","27798114630","12593387664","3748536792","726025872","88805574","6492200","257972","4680","26"],["22345512000","111840253980","243997528320","305206301400","242322400320","127771295652","45520290816","10943748816","1745295552","178358180","11019008","373464","5824","28"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","20","168","1344","10560","82368","640640","4978688","38697984","300984320","2343047168","18257510400","142408581120","1111882383360"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","2"],["-5","-7"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","0","1"],["2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180521,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-118d35e218f08ba2"]},{"id":180522,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-118d35e218f08ba2"]},{"id":180523,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-118d35e218f08ba2"]}],"region_exact":{},"incoming_edges":[{"id":"e-118d35e218f08ba2","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-16a684c04a1ed263","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":"2ee3452b10c64996031c574a21adc229f6a27e8860582b1be89407f1737c17e5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-25065a706ad43d32","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-88a79bc850a71581","kind":"fingerprint_related","node_a_id":"n-f14e2bec4affca4e","node_b_id":"n-16a684c04a1ed263","fingerprint":"f6-58fd21a6ca87df0d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-58fd21a6ca87df0d"},"other_id":"n-f14e2bec4affca4e","other":{"id":"n-f14e2bec4affca4e","name":"derivative of basis_map_A007318 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 peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-58fd21a6ca87df0d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["8","2"]],"card_rows":[["1"],["8","2"],["51","30","3"],["304","292","72","4"]],"row_sums_preview":["0","1","10","84","672","5280","41184","320320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A007318","D"],"edges":["e-38a7985c4c1aa2e3","e-d0cff2e14314e127"],"text":"A114656 \u2192 basis A007318 \u2192 D"},"headline":"A114656 \u2192 basis A007318 \u2192 D","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       0;\n   2:       2;\n   3:      16,        4;\n   4:     102,       60,        6;\n   5:     608,      584,      144,       8;\n   6:    3530,     4760,     1980,     280,      10;\n   7:   20256,    35340,    21120,    5160,     480,     12;\n   8:  115598,   248052,   195090,   69720,   11410,    756,    14;\n   9:  658048,  1678096,  1644384,  784112,  190400,  22512,  1120,   16;\n  10: 3742290, 11064240, 13015800, 7817040, 2561580, 454608, 40824, 1584, 18;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n-2)*P(n,x) = (5*n^2-5*n+2+(3*n^2-7*n+2)*x)*P(n-1,x) + (2*n-1-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 5*n + 2 + (3*n^2 - 7*n + 2)*x)*P[n - 1, x] + (2*n - 1 - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((5*n^2 - 5*n + 2 + (3*n^2 - 7*n + 2)*x)*v[i - 1] + (2*n - 1 - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a30765140629bd5","steps":["x \u21a6 x+1","D"],"edges":["e-5ebd2cd7e542e718","e-118d35e218f08ba2"],"text":"A114655 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A114655 \u2192 x \u21a6 x+1 \u2192 D","chain":null}
