{"id":"n-e21ecfeb95d562f5","revision":0,"first_index":0,"fingerprint":"f6-e63306eb47cd27c0","name":"derivative of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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":[[],["2"],["24","8"],["160","192","24"],["896","1920","960","64"],["4608","14336","13440","3840","160"],["22528","92160","129024","71680","13440","384"],["106496","540672","1013760","860160","322560","43008","896"],["491520","2981888","7028736","8110080","4730880","1290240","129024","2048"],["2228224","15728640","44728320","65601536","52715520","22708224","4730880","368640","4608"],["9961472","80216064","267386880","477102080","492011520","295206912","98402304","16220160","1013760","10240"],["44040192","398458880","1524105216","3208642560","4055367680","3148873728","1476034560","393609216","52715520","2703360","22528"],["192937984","1937768448","8367636480","20321402880","30482104320","29198647296","17843617792","6747586560","1476034560","164003840","7028736","49152"],["838860800","9261023232","44568674304","122725335040","213374730240","243856834560","184924766208","91767177216","28677242880","5248122880","492011520","17891328","106496"],["3623878656","43620761600","231525580800","713098788864","1411341352960","1877697626112","1706997841920","1056712949760","435894091776","114708971520","17843617792","1431306240","44728320","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","32","376","3840","36384","329216","2887552","24764416","208814592","1737531392","14304573440","116738818048","945752776704","7614541725696"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["3"],["-5"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","8"],["-12","-26"],["8","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","4"],["0","-4","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183272,"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-93b24cee7d491576"]},{"id":183273,"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-93b24cee7d491576"]},{"id":183274,"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-93b24cee7d491576"]},{"id":183275,"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-93b24cee7d491576"]}],"region_exact":{},"incoming_edges":[{"id":"e-93b24cee7d491576","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-e21ecfeb95d562f5","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":"4d62ace37c63571588f885ca4989e4ae3a71119e2e16c3ab5e39a0504ce9faf5","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-8cbd6010e0efad10","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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-dabceeb9a428b8ae","kind":"fingerprint_related","node_a_id":"n-fa59d237e613de64","node_b_id":"n-e21ecfeb95d562f5","fingerprint":"f6-e63306eb47cd27c0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e63306eb47cd27c0"},"other_id":"n-fa59d237e613de64","other":{"id":"n-fa59d237e613de64","name":"derivative of reversal of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","first_index":0,"fingerprint":"f6-e63306eb47cd27c0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["6","2"],["20","24","3"],["56","120","60","4"]],"card_rows":[["6","2"],["20","24","3"],["56","120","60","4"],["144","448","420","120","5"]],"row_sums_preview":["8","47","240","1137","5144","22559","96736","407841"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8eb9bb3a09f0b7fd","steps":["reverse","D"],"edges":["e-efd270083a78762c","e-0b01c87337cc5cbf"],"text":"A123519 \u2192 reverse \u2192 D"},"headline":"A123519 \u2192 reverse \u2192 D","chain":null}}],"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:       2;\n  2:      24,        8;\n  3:     160,      192,       24;\n  4:     896,     1920,      960,       64;\n  5:    4608,    14336,    13440,     3840,      160;\n  6:   22528,    92160,   129024,    71680,    13440,      384;\n  7:  106496,   540672,  1013760,   860160,   322560,    43008,     896;\n  8:  491520,  2981888,  7028736,  8110080,  4730880,  1290240,  129024,   2048;\n  9: 2228224, 15728640, 44728320, 65601536, 52715520, 22708224, 4730880, 368640, 4608;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (8*n^2-12*n+4+(12*n^2-26*n+8)*x)*P(n-1,x) - ((4*n-2)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 12*n + 4 + (12*n^2 - 26*n + 8)*x)*P[n - 1, x] - ((4*n - 2)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 12*n + 4 + (12*n^2 - 26*n + 8)*x)*v[i - 1] - ((4*n - 2)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3)); 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-c30cb06ad75c5e21","steps":["basis A013609","D"],"edges":["e-49766d0fffe14f49","e-93b24cee7d491576"],"text":"A103327 \u2192 basis A013609 \u2192 D"},"headline":"A103327 \u2192 basis A013609 \u2192 D","chain":null}