{"id":"n-814eff484ba2be47","revision":1,"first_index":1,"fingerprint":"f6-0ec5637665ef6d37","name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["12","8","1"],["50","54","15","1"],["210","320","150","24","1"],["882","1750","1190","330","35","1"],["3696","9072","8190","3360","630","48","1"],["15444","45276","51282","28350","7980","1092","63","1"],["64350","219648","300300","210672","80850","16800","1764","80","1"],["267410","1042470","1673100","1429428","702702","200970","32340","2700","99","1"],["1108536","4862000","8970390","9060480","5465460","2018016","450450","58080","3960","120","1"],["4585308","22355476","46650890","54478710","39090480","17825808","5171166","930930","98670","5610","143","1"],["18929092","101582208","236672436","314085200","261891630","142826112","51459408","12108096","1801800","160160","7722","168","1"],["78004500","457002364","1176307860","1749824076","1665113450","1060037550","458933904","134774640","26342316","3303300","250250","10374","195","1"],["320932800","2038517600","5746331500","9475362624","10143289156","7397630240","3753780030","1330243200","325909584","53909856","5785780","378560","13650","224","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","21","120","705","4188","24997","149488","894465","5351220","31997493","191193192","1141524033","6809904780","40592084805"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","4","-1"],["2","-6","1"],["-3","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["12","28","9","-5"],["-2","-24","-15","3"],["-10","-1","6","0"],["4","3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","7","2","-2"],["0","1","-3","-4","0"],["0","-2","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165674,"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-62274b400b6ee467"]},{"id":165675,"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-62274b400b6ee467"]},{"id":165676,"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-62274b400b6ee467"]},{"id":165677,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-62274b400b6ee467"]}],"region_exact":{},"incoming_edges":[{"id":"e-62274b400b6ee467","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-814eff484ba2be47","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":"033de22e451064fe8ed7635a00d2645f9c1aca790abed3ffd42d630664a78759","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-78ba05b4a3dd925d","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) 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":[],"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:      1;\n   2:      3,       1;\n   3:     12,       8,       1;\n   4:     50,      54,      15,       1;\n   5:    210,     320,     150,      24,      1;\n   6:    882,    1750,    1190,     330,     35,      1;\n   7:   3696,    9072,    8190,    3360,    630,     48,     1;\n   8:  15444,   45276,   51282,   28350,   7980,   1092,    63,    1;\n   9:  64350,  219648,  300300,  210672,  80850,  16800,  1764,   80,  1;\n  10: 267410, 1042470, 1673100, 1429428, 702702, 200970, 32340, 2700, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3-3*n^2+2*n+(2*n^2-6*n+4)*x+(n-1)*x^2)*P(n,x) = (4*n^3-10*n^2-2*n+12+(3*n^3-n^2-24*n+28)*x+(6*n^2-15*n+9)*x^2+(3*n-5)*x^3)*P(n-1,x) - ((2*n^2-n-3)*x+(2*n^2+3*n-7)*x^2+(4*n-2)*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 10*n^2 - 2*n + 12 + (3*n^3 - n^2 - 24*n + 28)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n - 5)*x^3)*P[n - 1, x] - ((2*n^2 - n - 3)*x + (2*n^2 + 3*n - 7)*x^2 + (4*n - 2)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n + (2*n^2 - 6*n + 4)*x + (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 10*n^2 - 2*n + 12 + (3*n^3 - n^2 - 24*n + 28)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n - 5)*x^3)*v[i - 1] - ((2*n^2 - n - 3)*x + (2*n^2 + 3*n - 7)*x^2 + (4*n - 2)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n + (2*n^2 - 6*n + 4)*x + (n - 1)*x^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-78ba05b4a3dd925d","steps":["D+1"],"edges":["e-62274b400b6ee467"],"text":"A104684 \u2192 D+1"},"headline":"A104684 \u2192 D+1","chain":null}