{"id":"n-c8e1d395ac5ec819","revision":1,"first_index":0,"fingerprint":"f6-01b9b2797a61b9d5","name":"derivative of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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"],["4","2"],["10","20","3"],["20","100","60","4"],["35","350","525","140","5"],["56","980","2940","1960","280","6"],["84","2352","12348","16464","5880","504","7"],["120","5040","42336","98784","70560","15120","840","8"],["165","9900","124740","465696","582120","249480","34650","1320","9"],["220","18150","326700","1829520","3659040","2744280","762300","72600","1980","10"],["286","31460","778635","6229080","18687240","22424688","10900890","2076360","141570","2860","11"],["364","52052","1717716","18894876","80978040","145760472","113369256","37789752","5153148","260260","4004","12"],["455","82810","3552549","52104052","307041735","789535890","921125205","491266776","117234117","11841830","455455","5460","13"],["560","127400","6956040","132628496","1042081040","3684500820","6140834700","4912667760","1875745872","331571240","25505480","764400","7280","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","6","33","184","1055","6222","37639","232808","1468080","9414800","61273080","403979952","2694246347","18153391102"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-12"],["5"],["6"],["1"]]},"terms":[{"coeff":{"coeffs":[["24","0"],["26","-10"],["9","0"],["1","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","24","12"],["0","33","3"],["0","3","-15"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","4"],["0","0","6","6"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164837,"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-d09a0797aedc5af0"]},{"id":164838,"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-d09a0797aedc5af0"]},{"id":164839,"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-d09a0797aedc5af0"]}],"region_exact":{},"incoming_edges":[{"id":"e-d09a0797aedc5af0","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-c8e1d395ac5ec819","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":"69d1f54a065a3f4230a773c7ee0715536013e45f3f4b4aa13ac4b7d75831caf0","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-047228a38014e6d4","role":"sequence","name":"Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","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:   0;\n  1:   1;\n  2:   4,    2;\n  3:  10,   20,      3;\n  4:  20,  100,     60,      4;\n  5:  35,  350,    525,    140,      5;\n  6:  56,  980,   2940,   1960,    280,      6;\n  7:  84, 2352,  12348,  16464,   5880,    504,     7;\n  8: 120, 5040,  42336,  98784,  70560,  15120,   840,    8;\n  9: 165, 9900, 124740, 465696, 582120, 249480, 34650, 1320, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2+5*n-12)*P(n,x) = (n^3+9*n^2+26*n+24+(10*n^3-10*n)*x)*P(n-1,x) + ((3*n^2+33*n+24)*x-(15*n^2-3*n-12)*x^2)*d/dx P(n-1,x) + ((6*n+4)*x^2+(6*n+4)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 - 10*n)*x)*P[n - 1, x] + ((3*n^2 + 33*n + 24)*x - (15*n^2 - 3*n - 12)*x^2)*D[P[n - 1, x], x] + ((6*n + 4)*x^2 + (6*n + 4)*x^3)*D[P[n - 1, x], {x, 2}])/(n^3 + 6*n^2 + 5*n - 12)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 9*n^2 + 26*n + 24 + (10*n^3 - 10*n)*x)*v[i - 1] + ((3*n^2 + 33*n + 24)*x - (15*n^2 - 3*n - 12)*x^2)*deriv(v[i - 1], x) + ((6*n + 4)*x^2 + (6*n + 4)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^3 + 6*n^2 + 5*n - 12)); 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-c963abfc0266d66c","steps":["\u2299 C(n+1,k+1) (shift +1)","D"],"edges":["e-548c39a476f786f5","e-d09a0797aedc5af0"],"text":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 D"},"headline":"A108838 \u2192 \u2299 C(n+1,k+1) (shift +1) \u2192 D","chain":null}